Part II

Bases and Frames

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Chapter 3Unconditional Convergence of Series in Banach and Hilbert Spaces[15/93%]
Problem 3.1

Given an orthonormal sequence {en}\left\{ e_{n}\right\} in a Hilbert space HH, prove that the following statements are equivalent.

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(a)

∑cnen\sum c_{n} e_{n} converges.

(b)

∑cnen\sum c_{n} e_{n} converges unconditionally.

(c)

∑∣cn∣2<∞\sum \left|c_{n}\right|^{2} < \infty.

Problem 3.2

Assume that ∑cn\sum c_{n} is a conditionally convergent series of real scalars, i.e., the series converges but does not converge unconditionally.

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(a)

Let (pn)\left(p_{n}\right) be the sequence of nonnegative terms of (cn)\left(c_{n}\right) in order, and let (qn)\left(q_{n}\right) be the sequence of negative terms of (cn)\left(c_{n}\right) in order. Show that ∑pn\sum p_{n} and ∑qn\sum q_{n} must both diverge.

(b)

Given x∈Rx \in \mathbb {R}, show there exists a permutation σ\sigma of N\mathbb {N} such that ∑cσ(n)\sum c_{\sigma (n)} converges and equals xx.

(c)

Show that there exists a permutation σ\sigma of N\mathbb {N} such that ∑cσ(n)\sum c_{\sigma (n)} diverges to ∞\infty, i.e., lim⁡N→∞∑n=1Ncσ(n)=∞\lim_{N \rightarrow \infty } \sum_{n = 1}^{N} c_{\sigma (n)} = \infty, and another permutation τ\tau such that ∑cτ(n)\sum c_{\tau (n)} diverges to −∞-\infty.

(d)

Show that there exists a permutation σ\sigma of N\mathbb {N} such that ∑cσ(n)\sum c_{\sigma (n)} does not converge and does not diverge to ∞\infty or −∞-\infty.

Problem 3.3

Let XX be a normed space. Prove that the following two statements are equivalent.

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(a)

XX is a Banach space.

(b)

Every absolutely convergent series in XX converges in XX. That is, if (xn)\left(x_{n}\right) is a sequence in XX and ∑∥xn∥<∞\sum \left\| x_{n}\right\| < \infty, then the series ∑xn\sum x_{n} converges in XX.

Problem 3.4

Given vectors xmnx_{m n} in a Banach space XX such that ∑m∑n∥xmn∥<∞\sum_{m} \sum_{n}\left\| x_{m n}\right\| < \infty, show that for any bijection σ:N→N×N\sigma : \mathbb {N} \rightarrow \mathbb {N} \times \mathbb {N} the following series exist and are equal:

∑m(∑nxmn)=∑n(∑mxmn)=∑kxσ(k) \sum _{m}\left(\sum _{n} x_{m n}\right) = \sum _{n}\left(\sum _{m} x_{m n}\right) = \sum _{k} x_{\sigma (k)}

Compare Theorem A.34, which formulates a similar result for series of scalars.

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Problem 3.5

Let XX be a Banach space, and fix A∈B(X)A \in \mathcal{B}(X). For n>0n > 0 let AnA^{n} denote the usual nnth power of AA (An=A⋯AA^{n} = A \cdots A, nn times), and define A0=IA^{0} = I (the identity map on XX).

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(a)

Given x∈Xx \in X, show that the series eA(x)=∑k=0∞Akxk!e^{A}(x) = \sum_{k = 0}^{\infty } \frac{A^{k} x}{k!} converges absolutely in XX, and show that eAe^{A} is a linear operator on XX.

(b)

Prove that the series ∑k=0∞Akk!\sum_{k = 0}^{\infty } \frac{A^{k}}{k!} converges absolutely in B(X)\mathcal{B}(X), and equals the operator eAe^{A} defined in part (a). Conclude that eA∈B(X)e^{A} \in \mathcal{B}(X) and ∥eA∥≤e∥A∥\left\| e^{A}\right\| \leq e^{\left\| A\right\| }.

(c)

Prove that if A,B∈B(X)A, B \in \mathcal{B}(X) and AB=BAA B = B A, then eAeB=eA+B=eBeAe^{A} e^{B} = e^{A+B} = e^{B} e^{A}.

(d)

Let HH be a Hilbert space. Show that if A∈B(H)A \in \mathcal{B}(H) is self-adjoint, then eiAe^{i A} is unitary.

Problem 3.6

Let HH be a (possibly nonseparable) Hilbert space and let II be a (possibly uncountable) index set. Show that if {xi}i∈I\left\{ x_{i}\right\}_{i \in I} is an orthonormal set in HH, then the following statements hold.

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(a)

If x∈Hx \in H then ⟨x,xi⟩≠0\left\langle x, x_{i} \right\rangle \neq 0 for at most countably many i∈Ii \in I.

(b)

For each x∈H,∑i∈I∣⟨x,xi⟩∣2≤∥x∥2x \in H, \sum_{i \in I}\left| \left\langle x, x_{i} \right\rangle \right|^{2} \leq \left\| x\right\|^{2}.

(c)

For each x∈Hx \in H, the series p=∑i∈I⟨x,xi⟩xip = \sum_{i \in I} \left\langle x, x_{i} \right\rangle x_{i} converges with respect to the net of finite subsets of II, and pp is the orthogonal projection of xx onto span⁡‾{xi}i∈I\overline{\operatorname {span}}\left\{ x_{i}\right\}_{i \in I}.

Problem 3.7

Let HH be a (possibly nonseparable) Hilbert space and let II be an index set. Given an orthonormal set {xi}i∈I\left\{ x_{i}\right\}_{i \in I} in HH, prove that the following statements are equivalent.

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(a)

{xi}i∈I\left\{ x_{i}\right\}_{i \in I} is complete.

(b)

For each x∈Hx \in H we have x=∑i∈I⟨x,xi⟩xix = \sum_{i \in I} \left\langle x, x_{i} \right\rangle x_{i}, where the series converges with respect to the net of finite subsets of II.

(c)

For each x∈H,∥x∥2=∑i∈I∣⟨x,xi⟩∣2x \in H,\left\| x\right\|^{2} = \sum_{i \in I}\left| \left\langle x, x_{i} \right\rangle \right|^{2}.

Problem 3.8

Let {xn}\left\{ x_{n}\right\} be a sequence in a Banach space XX. Fix 1≤p≤∞1 \leq p \leq \infty, and suppose that T(μ)=(⟨xn,μ⟩)∈ℓp′T(\mu ) = \left( \left\langle x_{n}, \mu \right\rangle \right) \in \ell^{p^{\prime }} for every μ∈X∗\mu \in X^{*}. Prove the following statements about TT, which is called the analysis operator associated with {xn}\left\{ x_{n}\right\}. These facts will be useful to us in Chapters 7 and 8.

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(a)

T:X∗→ℓp′T: X^{*} \rightarrow \ell^{p^{\prime }} is bounded and linear.

(b)

If 1≤p<∞1 \leq p < \infty then the series ∑cnxn\sum c_{n} x_{n} converges unconditionally for each (cn)∈ℓp\left(c_{n}\right) \in \ell^{p}, and the synthesis operator Uc=∑cnxnU c = \sum c_{n} x_{n} is a bounded map of ℓp\ell^{p} into XX.

(c)

If p=∞p = \infty and (cn)∈ℓ1\left(c_{n}\right) \in \ell^{1}, then the series ∑cnxn\sum c_{n} x_{n} converges weakly, i.e., ∑cn⟨xn,μ⟩\sum c_{n} \left\langle x_{n}, \mu \right\rangle converges for each μ∈X∗\mu \in X^{*}. However, ∑cnxn\sum c_{n} x_{n} need not converge in the norm of XX.

(d)

If 1≤p<∞1 \leq p < \infty then T=U∗T = U^{*}.

(e)

If 1<p<∞1 < p < \infty and XX is reflexive, then U=T∗U = T^{*}.

Problem 3.9
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(a)

Let X=ℓ∞X = \ell^{\infty }, and let {δn}\left\{ \delta_{n}\right\} be the sequence of standard basis vectors (which is contained in ℓ∞\ell^{\infty } but does not form a basis for it). Show that R=RE=RΛ=1\mathcal{R} = \mathcal{R}_{\mathcal{E}} = \mathcal{R}_{\Lambda } = 1, but ∑δn\sum \delta_{n} does not converge in ℓ∞\ell^{\infty }.

(b)

Exhibit a sequence {xn}\left\{ x_{n}\right\} in a separable Banach space XX such that R\mathcal{R}, RΛ\mathcal{R}_{\Lambda }, and RE\mathcal{R}_{\mathcal{E}} are finite but ∑xn\sum x_{n} does not converge.

Problem 3.10

Suppose that {xn}\left\{ x_{n}\right\} is a sequence in a Hilbert space HH that is bounded above and below in norm. Show that if ∑cnxn\sum c_{n} x_{n} converges unconditionally, then (cn)∈ℓ2\left(c_{n}\right) \in \ell^{2}. Is the converse true?

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Problem 3.11

Prove Lemma 3.26 for the case of complex scalars.

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Problem 3.12

Prove Theorem 3.28.

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Remark.
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Theorem 3.28: Fix 1≤p≤21 \leq p \leq 2, and let E⊆RE \subseteq \mathbb {R} be a measurable set with ∣E∣<∞\left|E\right| < \infty. Let {fn}\left\{ f_{n}\right\} be an orthonormal basis for L2(E)L^{2}(E) such that A=inf⁡n∥fn∥Lp>0A = \inf_{n}\left\| f_{n}\right\|_{L^{p}} > 0. If f=∑ncnfnf = \sum_{n} c_{n} f_{n} converges unconditionally in Lp(E)L^{p}(E), then ∑∣cn∣2<∞\sum \left|c_{n}\right|^{2} < \infty and f∈L2(E)f \in L^{2}(E).

Problem 3.13

Prove the following version of Orlicz's Theorem for the case 2<p<∞2 < p < \infty : If ∑fn\sum f_{n} converges unconditionally in Lp(E)L^{p}(E) then ∑∥fn∥Lpp<∞\sum \left\| f_{n}\right\|_{L^{p}}^{p} < \infty.

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Problem 3.14

Let XX be a finite-dimensional normed space. Show that a series ∑xn\sum x_{n} in XX is unconditionally convergent if and only if it is absolutely convergent.

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Problem 3.15

Let F=R\mathbf{F} = \mathbb {R}. Show that C[0,1]C[0,1] is a Banach lattice with respect to the ordering ≤\leq given in equation (3.13). Also show that f∨g=max⁡{f,g},f∧g=min⁡{f,g}f \vee g = \max \left\{ f, g\right\} , f \wedge g = \min \left\{ f, g\right\}, and ∣f∣\left|f\right| is the ordinary absolute value of ff.

Remark Another example of a Banach lattice is given in Exercise 6.6.

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Chapter 4Bases in Banach Spaces[29/100%]
Problem 4.1

Use Zorn's Lemma to show that if VV is a vector space then there exists a sequence {xi}i∈I\left\{ x_{i}\right\}_{i \in I} that is a Hamel basis for VV.

Remark Although we focus in this volume on real and complex vector spaces, the argument of this exercise applies to any vector space VV over any field F\mathbf{F}.

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Problem 4.2

Let XX be an infinite-dimensional Banach space, and prove the following statements.

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(a)

Any Hamel basis for XX must be uncountable.

(b)

Any infinite-dimensional subspace of XX that has a countable Hamel basis is a meager subset of XX, and cannot be a closed subspace of XX.

(c)

By Exercise 2.26, Cc(R)C_{c}(\mathbb {R}) is a meager, dense subspace of C0(R)C_{0}(\mathbb {R}). Show that Cc(R)C_{c}(\mathbb {R}) does not have a countable Hamel basis.

Problem 4.3

Let {xi}i∈I\left\{ x_{i}\right\}_{i \in I} be a Hamel basis for an infinite-dimensional Banach space XX. Then each x∈Xx \in X can be written uniquely as x=∑i∈Iai(x)xix = \sum_{i \in I} a_{i}(x) x_{i} where at most finitely many of the scalars ai(x)a_{i}(x) are nonzero. Each aia_{i} is a linear functional on XX, and we call {ai}i∈I\left\{ a_{i}\right\}_{i \in I} the sequence of coefficient functionals associated with the Hamel basis {xi}i∈I\left\{ x_{i}\right\}_{i \in I}. This exercise addresses the question of whether these coefficient functionals can be continuous.

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(a)

Show by example that it is possible for some particular functional aia_{i} to be continuous.

(b)

Show that ai(xj)=δija_{i}\left(x_{j}\right) = \delta_{i j} for i,j∈Ii, j \in I, where δij\delta_{i j} is the Kronecker delta.

(c)

Let J={i∈I:ai is continuous}J = \left\{ i \in I: a_{i} \text{ is continuous}\right\}. Show that sup⁡j∈J∥aj∥<∞\sup_{j \in J}\left\| a_{j}\right\| < \infty.

(d)

Show that at most finitely many functionals aia_{i} can be continuous, i.e., JJ is finite.

(e)

Give an example of an infinite-dimensional normed linear space that has a Hamel basis {xi}i∈I\left\{ x_{i}\right\}_{i \in I} such that each of the associated coefficient functionals aia_{i} for i∈Ii \in I is continuous.

Problem 4.4

Let μ\mu be an unbounded linear functional on an infinite-dimensional Banach space XX (see Example 4.2). Then by Exercise 1.16, X1=X×FX_{1} = X \times \mathbf{F} is a Banach space with respect to the norm ∥(x,c)∥X1=∥x∥X+∣c∣\left\| (x, c)\right\|_{X_{1}} = \left\| x\right\|_{X}+\left|c\right|. Set Y=graph⁡(μ)={(x,μ(x)):x∈X}Y = \operatorname {graph}(\mu ) = \left\{ (x, \mu (x)): x \in X\right\}, and define ∥(x,μ(x))∥Y=∥x∥X\left\| (x, \mu (x))\right\|_{Y} = \left\| x\right\|_{X}.

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(a)

Show that (Y,∥⋅∥Y)(Y,\left\| \cdot \right\|_{Y}) is a Banach space.

(b)

Show that even though Y⊆X1Y \subseteq X_{1}, the normed space (Y,∥⋅∥Y)(Y,\left\| \cdot \right\|_{Y}) is not continuously embedded into (X1,∥⋅∥X1)(X_{1},\left\| \cdot \right\|_{X_{1}}), i.e., the mapping I:(Y,∥⋅∥Y)→(X1,∥⋅∥X1)I:\left(Y,\left\| \cdot \right\|_{Y}\right) \rightarrow (X_{1},\left\| \cdot \right\|_{X_{1}}) given by I(z)=zI(z) = z is not continuous.

Problem 4.5

Since the set Q\mathbb {Q} of rational numbers is a field, we can consider the vector space R\mathbb {R} over the field Q\mathbb {Q}. By Exercise 4.1, there exists a Hamel basis {xi}i∈I\left\{ x_{i}\right\}_{i \in I} for R\mathbb {R} over Q\mathbb {Q}. Hence every nonzero number x∈Rx \in \mathbb {R} can be written uniquely as x=∑k=1Nckxikx = \sum_{k = 1}^{N} c_{k} x_{i_{k}} for some i1,…,iN∈Ii_{1}, \ldots , i_{N} \in I and nonzero rational scalars c1,…,cNc_{1}, \ldots , c_{N}. Use this to show that there exists a function f:R→Rf: \mathbb {R} \rightarrow \mathbb {R} that satisfies f(x+y)=f(x)+f(y)f(x+y) = f(x)+f(y) for all x,y∈Rx, y \in \mathbb {R} but does not satisfy f(cx)=cf(x)f(c x) = c f(x) for all c,x∈Rc, x \in \mathbb {R}. Thus ff is not linear, even though ff respects addition.

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Problem 4.6

Let {xn}\left\{ x_{n}\right\} be a basis for a Banach space XX, and let (λn)\left(\lambda_{n}\right) be a sequence of nonzero scalars.

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(a)

Show that {λnxn}\left\{ \lambda_{n} x_{n}\right\} is a basis for XX. In particular, {xn/∥xn∥}\left\{ x_{n} /\left\| x_{n}\right\| \right\} is a normalized basis for XX.

(b)

If {xn}\left\{ x_{n}\right\} is an unconditional basis, will {λnxn}\left\{ \lambda_{n} x_{n}\right\} be an unconditional basis?

(c)

If {xn}\left\{ x_{n}\right\} is an absolutely convergent basis, will {λnxn}\left\{ \lambda_{n} x_{n}\right\} be an absolutely convergent basis?

Problem 4.7

Let HH be a separable Hilbert space.

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(a)

Show that an orthonormal basis for HH is a normalized unconditional basis for HH.

(b)

Give an example of a basis {xn}\left\{ x_{n}\right\} for HH that is not an orthogonal sequence.

(c)

Give an example of a basis {xn}\left\{ x_{n}\right\} for HH that contains no orthogonal subsequences (equivalently, ⟨xm,xn⟩≠0\left\langle x_{m}, x_{n} \right\rangle \neq 0 for all m≠nm \neq n).

Problem 4.8
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(a)

Show that the standard basis {δn}n∈N\left\{ \delta_{n}\right\}_{n \in \mathbb {N}} is a normalized unconditional basis for ℓp\ell^{p} for each 1≤p<∞1 \leq p < \infty, and is also a normalized unconditional basis for c0c_{0}.

(b)

By Exercise 1.20, c={x=(xn)∈ℓ∞:lim⁡n→∞xn exists}c = \left\{ x = \left(x_{n}\right) \in \ell^{\infty }: \lim_{n \rightarrow \infty } x_{n} \text{ exists}\right\} is a closed subspace of ℓ∞\ell^{\infty }, and c0c_{0} is a proper closed subspace of cc. Find a vector δ0∈c\delta_{0} \in c such that {δn}n≥0\left\{ \delta_{n}\right\}_{n \geq 0} is a normalized unconditional basis for cc.

(c)

Show that c∗c^{*} is isometrically isomorphic to ℓ1\ell^{1}. Compare Exercise 1.75, which shows that we also have c0∗≅ℓ1c_{0}{ }^{*} \cong \ell^{1}, and Exercise 4.22, which shows that cc and c0c_{0} are topologically isomorphic but not isometrically isomorphic.

Problem 4.9

For each n∈Nn \in \mathbb {N}, define yn=(1,…,1,0,0,…)y_{n} = (1, \ldots , 1,0,0, \ldots ), where the 1 is repeated nn times. Show that {yn}\left\{ y_{n}\right\} is a normalized conditional basis for c0c_{0}.

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Problem 4.10

For each n∈Nn \in \mathbb {N}, define zn=(0,…,0,1,1,…)z_{n} = (0, \ldots , 0,1,1, \ldots ), where the 0 is repeated n−1n-1 times. Show that {zn}\left\{ z_{n}\right\} is a normalized conditional basis for cc (this is called the summing basis for cc).

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Problem 4.11

Show that the standard basis {δn}n∈N\left\{ \delta_{n}\right\}_{n \in \mathbb {N}} is a monotone basis for ℓp\ell^{p} for each index 1≤p<∞1 \leq p < \infty, and is also a monotone basis for c0c_{0}. Is the basis {δn}n≥0\left\{ \delta_{n}\right\}_{n \geq 0} for cc given in Exercise 4.8 monotone?

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Problem 4.12

Show that any subsequence of a basis is a basic sequence.

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Problem 4.13

Suppose that {xn}\left\{ x_{n}\right\} is a Schauder basis for a Hilbert space HH, and its biorthogonal system is {yn}\left\{ y_{n}\right\}. Show that if ∥xn∥=∥yn∥=1\left\| x_{n}\right\| = \left\| y_{n}\right\| = 1 for every nn, then {xn}\left\{ x_{n}\right\} is an orthonormal basis for HH.

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Problem 4.14
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(a)

Show that if {xn}\left\{ x_{n}\right\} is an absolutely convergent basis for a Banach space XX, then

Ty=∑n=1∞yn∥xn∥xn,y=(yn)∈ℓ1 T y = \sum _{n = 1}^{\infty } \frac{y_{n}}{\left\| x_{n}\right\| } x_{n}, \quad y = \left(y_{n}\right) \in \ell ^{1}

is a topological isomorphism of ℓ1\ell^{1} onto XX.

(b)

Conversely, show that every Banach space topologically isomorphic to ℓ1\ell^{1} has an absolutely convergent basis.

Problem 4.15

This exercise will give an alternative approach to proving Theorem 4.13. Let {xn}\left\{ x_{n}\right\} be a basis for a Banach space XX, and set ∥x∥=sup⁡N∥SNx∥\left\| x\right\| = \sup_{N}\left\| S_{N} x\right\|, where the SNS_{N} are the partial sum operators.

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(a)

Show that ∣∣∣⋅∣∣∣|||\cdot ||| is a norm on XX, and ∥x∥≤∥x∥\left\| x\right\| \leq \left\| x\right\| for all x∈Xx \in X.

(b)

Suppose that {yn}\left\{ y_{n}\right\} is a Cauchy sequence in XX with respect to ∣∣∣⋅∣∣∣|||\cdot |||. With NN fixed, show that {SNyn}n∈N\left\{ S_{N} y_{n}\right\}_{n \in \mathbb {N}} is Cauchy with respect to ∥⋅∥\left\| \cdot \right\|. Let zNz_{N} be such that ∥zN−SNyn∥→0\left\| z_{N}-S_{N} y_{n}\right\| \rightarrow 0 as n→∞n \rightarrow \infty, and observe that zN∈span⁡{x1,…,xN}z_{N} \in \operatorname {span}\left\{ x_{1}, \ldots , x_{N}\right\}.

(c)

Show that for each N∈NN \in \mathbb {N} we have lim⁡n→∞(sup⁡N∥zN−SNyn∥)=0\lim_{n \rightarrow \infty }\left(\sup_{N}\left\| z_{N}-S_{N} y_{n}\right\| \right) = 0, and use this to show that {zN}N∈N\left\{ z_{N}\right\}_{N \in \mathbb {N}} is Cauchy with respect to ∥⋅∥\left\| \cdot \right\|. Let y∈Xy \in X be the element such that ∥y−zN∥→0\left\| y-z_{N}\right\| \rightarrow 0.

(d)

Show that SN(zN+1)=zNS_{N}\left(z_{N+1}\right) = z_{N}, and use this to show that zN=∑n=1Ncnxnz_{N} = \sum_{n = 1}^{N} c_{n} x_{n} where cnc_{n} is independent of NN.

(e)

Show that y=∑n=1∞cnxny = \sum_{n = 1}^{\infty } c_{n} x_{n}, and hence zN=SNyz_{N} = S_{N} y. Use this to show that ∥y−yn∥→0\left\| y-y_{n}\right\| \rightarrow 0, and conclude that XX is complete with respect to ∣∣∣⋅∣∣∣|||\cdot |||.

(f)

Show that ∥⋅∥\left\| \cdot \right\| and ∣∣∣⋅∣∣∣|||\cdot ||| are equivalent norms on XX, and use this to show that C=sup⁡N∥SN∥<∞\mathcal{C} = \sup_{N}\left\| S_{N}\right\| < \infty. Conclude that {xn}\left\{ x_{n}\right\} is a Schauder basis for XX.

Problem 4.16

We say that a Banach space XX has the approximation property if the identity operator on XX can be uniformly approximated on every compact subset of XX by operators with finite rank. That is, given a compact set K⊆XK \subseteq X, there must exist continuous finite rank operators TNT_{N} such that

lim⁡N→∞(sup⁡x∈K∥x−TNx∥)=0. \lim _{N \rightarrow \infty }\left(\sup _{x \in K}\left\| x-T_{N} x\right\| \right) = 0.
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(a)

Show that if XX has a basis, then XX has the approximation property.

(b)

Suppose that YY is a Banach space that has the approximation property and XX is an arbitrary Banach space. Show that if T:X→YT: X \rightarrow Y is a compact operator, then there exist continuous finite-rank operators TN:X→YT_{N}: X \rightarrow Y such that ∥T−TN∥→0\left\| T-T_{N}\right\| \rightarrow 0 as N→∞N \rightarrow \infty.

Remark Compact sets and compact operators are reviewed in Appendix B.

Problem 4.17

Let XX be a Banach space.

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(a)

Show that if {xn}\left\{ x_{n}\right\} is a basis for XX, then the only possible vector y∈Xy \in X such that xn→wyx_{n} \xrightarrow {\mathrm{w}} y is y=0y = 0 (weak convergence is defined in Section 2.10).

(b)

Use Theorem 2.39 to show that part (a) remains valid if we only assume that {xn}\left\{ x_{n}\right\} is a basic sequence in XX.

(c)

Give an example of a basis {xn}\left\{ x_{n}\right\} for a Banach space XX such that xn→w0x_{n} \xrightarrow {\mathrm{w}} 0.

(d)

Give an example of a basis {xn}\left\{ x_{n}\right\} for a Banach space XX such that {xn}\left\{ x_{n}\right\} does not converge weakly to any vector in XX.

Problem 4.18

Prove the implication (a) ⇒\Rightarrow (b) in Theorem 4.20.

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Remark.
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Theorem 4.20: Let XX and YY be Banach spaces. If {xn}\left\{ x_{n}\right\} is a basis for XX and {yn}\left\{ y_{n}\right\} is a basis for YY, then the following two statements are equivalent.

  1. {xn}\left\{ x_{n}\right\} is equivalent to {yn}\left\{ y_{n}\right\}.

  2. ∑cnxn\sum c_{n} x_{n} converges in XX if and only if ∑cnyn\sum c_{n} y_{n} converges in YY. Definition 4.19: A basis {xn}\left\{ x_{n}\right\} for XX is equivalent to a basis {yn}\left\{ y_{n}\right\} for YY if there exists a topological isomorphism T:X→YT: X \rightarrow Y such that Txn=ynT x_{n} = y_{n} for all nn.

Problem 4.19

Suppose that {xn}\left\{ x_{n}\right\} is a basis for a Banach space X,{yn}X,\left\{ y_{n}\right\} is a sequence in a normed space YY, and there exists a topological isomorphism T:X→YT: X \rightarrow Y such that Txn=ynT x_{n} = y_{n} for every nn. Show that YY is a Banach space, {yn}\left\{ y_{n}\right\} is a basis for YY, and {xn}\left\{ x_{n}\right\} is equivalent to {yn}\left\{ y_{n}\right\}.

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Problem 4.20

Suppose {xn}\left\{ x_{n}\right\} is a basis for a Banach space XX that is equivalent to a basis {yn}\left\{ y_{n}\right\} for a Banach space YY.

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(a)

Show that {xn}\left\{ x_{n}\right\} is a bounded basis for XX if and only if {yn}\left\{ y_{n}\right\} is a bounded basis for YY.

(b)

Show that {xn}\left\{ x_{n}\right\} is an unconditional basis for XX if and only if {yn}\left\{ y_{n}\right\} is an unconditional basis for YY.

(c)

Show that {xn}\left\{ x_{n}\right\} is an absolutely convergent basis for XX if and only if {yn}\left\{ y_{n}\right\} is an absolutely convergent basis for YY.

Problem 4.21

Let {xn}\left\{ x_{n}\right\} be a basis for a Banach space XX and {yn}\left\{ y_{n}\right\} a basis for a Banach space YY. Show that {xn}\left\{ x_{n}\right\} is equivalent to {yn}\left\{ y_{n}\right\} if and only if there exist constants C1,C2>0C_{1}, C_{2} > 0 such that for all N∈NN \in \mathbb {N} and c1,…,cN∈Fc_{1}, \ldots , c_{N} \in \mathbf{F} we have

C1∥∑n=1Ncnyn∥≤∥∑n=1Ncnxn∥≤C2∥∑n=1Ncnyn∥. C_{1}\left\| \sum _{n = 1}^{N} c_{n} y_{n}\right\| \leq \left\| \sum _{n = 1}^{N} c_{n} x_{n}\right\| \leq C_{2}\left\| \sum _{n = 1}^{N} c_{n} y_{n}\right\| .
?
Problem 4.22
?
(a)

Let {δn}\left\{ \delta_{n}\right\} be the standard basis for c0c_{0}. By Exercise 4.8, if we set δ0=(1,1,…)\delta_{0} = (1,1, \ldots ), then {δn}n≥0\left\{ \delta_{n}\right\}_{n \geq 0} is a basis for cc. Show that cc and c0c_{0} are topologically isomorphic, and these two bases are equivalent.

(b)

Show that if x∈c0x \in c_{0} and ∥x∥∞=1\left\| x\right\|_{\infty } = 1, then there exist y≠z∈c0y \neq z \in c_{0} with ∥y∥∞=∥z∥∞=1\left\| y\right\|_{\infty } = \left\| z\right\|_{\infty } = 1 such that x=(y+z)/2x = (y+z) / 2. Show that the analogous statement for cc can fail.

(c)

Show that cc is not isometrically isomorphic to c0c_{0} (even so, note that their dual spaces c∗c^{*} and c0∗c_{0}{ }^{*} are each isometrically isomorphic to ℓ1\ell^{1} by Exercises 1.75 and 4.8).

Problem 4.23

Prove that the representation of functions f∈C[0,1]f \in C[0,1] given in equation (4.5) is unique.

?
Problem 4.24
?
(a)

Prove that C(T)C(\mathbf{T}) is a Banach space with respect to the uniform norm.

(b)

Show that Cper [0,1]C_{\text{per }}[0,1] is a proper closed subspace of C[0,1]C[0,1], and that Cper [0,1]C_{\text{per }}[0,1] is isometrically isomorphic to C(T)C(\mathbf{T}).

(c)

Show that Lp(T)L^{p}(\mathbf{T}) is isometrically isomorphic to Lp[0,1]L^{p}[0,1] for each 1≤p≤∞1 \leq p \leq \infty.

(d)

Show that if 1≤p<q≤∞1 \leq p < q \leq \infty, then Lq(T)⊊Lp(T)L^{q}(\mathbf{T}) \subsetneq L^{p}(\mathbf{T}).

Problem 4.25

Prove Theorem 4.27.

?
Remark.
?

Theorem 4.27: Let XX be a Banach space. If {xn}\left\{ x_{n}\right\} is a strong basis for XX, then {xn}\left\{ x_{n}\right\} is a weak basis for XX. Further, in this case {xn}\left\{ x_{n}\right\} is a weak Schauder basis for XX with coefficient functionals that are strongly continuous on XX.

Problem 4.26

Prove Theorem 4.28: let {xn}\left\{ x_{n}\right\} be a sequence in a Banach space XX, and let Y={(cn):∑cnxn converges weakly in X}Y = \left\{ \left(c_{n}\right): \sum c_{n} x_{n}\text{ converges weakly in }X\right\} with norm ∥(cn)∥Y=sup⁡N∥∑n=1Ncnxn∥\left\| \left(c_{n}\right)\right\|_{Y} = \sup_{N}\left\| \sum_{n = 1}^{N} c_{n} x_{n}\right\|.

?
(a)

YY is a Banach space.

(b)

If {xn}\left\{ x_{n}\right\} is a weak basis for XX, then YY is topologically isomorphic to XX via (cn)↦∑cnxn\left(c_{n}\right) \mapsto \sum c_{n} x_{n}.

Problem 4.27

Finish the details of Example 4.31.

?
Remark.
?

Example 4.31: Let X=ℓ1X = \ell^{1}, and note that X∗=ℓ∞X^{*} = \ell^{\infty } is not separable. The sequence of standard basis vectors {δn}\left\{ \delta_{n}\right\} is contained in ℓ∞\ell^{\infty }, although it does not form a strong basis for this space. We will show that {δn}\left\{ \delta_{n}\right\} is a weak* basis for ℓ∞\ell^{\infty }. Choose any y=(yn)∈ℓ∞y = \left(y_{n}\right) \in \ell^{\infty }. Then for any x=(xn)∈ℓ1x = \left(x_{n}\right) \in \ell^{1} we have

lim⁡N→∞⟨x,∑n=1Nynδn⟩=lim⁡N→∞∑n=1N⟨x,δn⟩yn=lim⁡N→∞∑n=1Nxnyn=⟨x,y⟩. \lim _{N \rightarrow \infty } \left\langle x, \sum _{n = 1}^{N} y_{n} \delta _{n} \right\rangle = \lim _{N \rightarrow \infty } \sum _{n = 1}^{N} \left\langle x, \delta _{n} \right\rangle y_{n} = \lim _{N \rightarrow \infty } \sum _{n = 1}^{N} x_{n} y_{n} = \left\langle x, y \right\rangle .

Hence y=∑ynδny = \sum y_{n} \delta_{n} in the weak* topology (even though this series need not converge strongly), and by Exercise 4.27 this representation is unique. Therefore {δn}\left\{ \delta_{n}\right\} is a weak* basis for ℓ∞\ell^{\infty }.

Here a sequence {xn∗}\left\{ x_{n}^{*}\right\} in X∗X^{*} is a weak* basis for X∗X^{*} if for each x∗∈X∗x^{*} \in X^{*} there exist unique scalars an∗(x∗)a_{n}^{*}\left(x^{*}\right) such that x∗=∑an∗(x∗)xn∗x^{*} = \sum a_{n}^{*}\left(x^{*}\right) x_{n}^{*}, with convergence of this series in the weak* topology.

Problem 4.28

Prove the statements made in Example 4.32:

?
(a)

{xn}⊆ℓ1\left\{ x_{n}\right\} \subseteq \ell^{1} is a strong Schauder basis for ℓ1\ell^{1} whose biorthogonal system is {yn}\left\{ y_{n}\right\}.

(b)

The representation x=∑⟨x,yn⟩xnx = \sum \left\langle x, y_{n} \right\rangle x_{n} is the unique representation of xx with respect to weak* convergence, i.e., {xn}\left\{ x_{n}\right\} is a weak* basis for ℓ1\ell^{1}.

(c)

y1∉c0y_{1} \notin c_{0}, so the functional determined by y1y_{1} is not weak*-continuous on ℓ1\ell^{1}; hence {xn}\left\{ x_{n}\right\} is not a weak* Schauder basis for ℓ1\ell^{1}.

Problem 4.29

Prove the statements made in Example 4.33:

?
(a)

{xn}\left\{ x_{n}\right\} is a strong Schauder basis for ℓ1\ell^{1} whose biorthogonal system is {yn}\left\{ y_{n}\right\}.

(b)

∑xn=0\sum x_{n} = 0 with weak* convergence, so {xn}\left\{ x_{n}\right\} is not weak* ω\omega-independent; hence x=∑⟨x,yn⟩xnx = \sum \left\langle x, y_{n} \right\rangle x_{n} is not the unique weak*-convergent representation of xx, so {xn}\left\{ x_{n}\right\} is not a weak* basis for ℓ1\ell^{1}.

Chapter 5Biorthogonality, Minimality, and More About Bases[21/100%]
Problem 5.1

Prove part (b) of Lemma 5.4.

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Remark.
?

Lemma 5.4: Let {xn}\left\{ x_{n}\right\} be a sequence in a Banach space XX.

  1. ∃{an}⊆X∗\exists \left\{ a_{n}\right\} \subseteq X^{*} biorthogonal to {xn}\left\{ x_{n}\right\}   ⟺  \iff {xn}\left\{ x_{n}\right\} is minimal.

  2. ∃\exists unique {an}⊆X∗\left\{ a_{n}\right\} \subseteq X^{*} biorthogonal to {xn}\left\{ x_{n}\right\}   ⟺  \iff {xn}\left\{ x_{n}\right\} is exact. Definition 5.3: A sequence {xn}\left\{ x_{n}\right\} in a Banach space XX is minimal if no vector xmx_{m} lies in the closed span of the other vectors xnx_{n}, i.e., xm∉span⁡‾{xn}n≠mx_{m} \notin \overline{\operatorname {span}}\left\{ x_{n}\right\}_{n \neq m} for all m∈Nm \in \mathbb {N}. A sequence that is both minimal and complete is said to be exact. Here {an}\left\{ a_{n}\right\} is biorthogonal to {xn}\left\{ x_{n}\right\} if ⟨xm,an⟩=δmn\left\langle x_{m}, a_{n} \right\rangle = \delta_{m n} for all m,nm, n.

Problem 5.2

Fix 1≤p<∞1 \leq p < \infty.

?
(a)

Show that {xk}k≥0\left\{ x^{k}\right\}_{k \geq 0} is complete in Lp[0,1]L^{p}[0,1].

(b)

Show that {xk}k≥0\left\{ x^{k}\right\}_{k \geq 0} is not minimal in Lp[0,1]L^{p}[0,1].

(c)

Show that {xk}k≥0\left\{ x^{k}\right\}_{k \geq 0} is not complete in L∞[0,1]L^{\infty }[0,1].

Problem 5.3

Let {en}\left\{ e_{n}\right\} be an orthonormal basis for a Hilbert space HH. Show that each of the following two sequences are ω\omega-independent and complete in HH, but are not minimal.

?
(a)

{e1}∪{en+en+1}n∈N\left\{ e_{1}\right\} \cup \left\{ e_{n}+e_{n+1}\right\}_{n \in \mathbb {N}}.

(b)

{e1}∪{e1+1nen}n≥2\left\{ e_{1}\right\} \cup \left\{ e_{1}+\frac{1}{n} e_{n}\right\}_{n \geq 2}.

Problem 5.4

Let {en}\left\{ e_{n}\right\} be an orthonormal basis for a Hilbert space HH, and for each n∈Nn \in \mathbb {N} define xn=∑k=1nek/kx_{n} = \sum_{k = 1}^{n} e_{k} / k. Show that {xn}\left\{ x_{n}\right\} is minimal and complete in HH, but is not a Schauder basis for HH.

?
Problem 5.5

Let {xn}\left\{ x_{n}\right\} be a sequence in a Banach space XX. Show that if {xn}\left\{ x_{n}\right\} is ω\omega independent and for every x∈Xx \in X there exist some scalars (cn)\left(c_{n}\right) such that x=∑cnxnx = \sum c_{n} x_{n}, then {xn}\left\{ x_{n}\right\} is a basis for XX.

?
Problem 5.6

Let XX be a Banach space. Let us say that a sequence {PN}⊆B(X)\left\{ P_{N}\right\} \subseteq \mathcal{B}(X) is a family of partial sum projections if: (i) PN2=PNP_{N}^{2} = P_{N}, (ii) PNx→xP_{N} x \rightarrow x as N→∞N \rightarrow \infty for each x∈Xx \in X, (iii) dim⁡(range⁡(PN))=N\operatorname {dim}\left(\operatorname {range}\left(P_{N}\right)\right) = N, and (iv) PNPM=Pmin⁡{M,N}P_{N} P_{M} = P_{\min \left\{ M, N\right\} }.

?
(a)

Show that if {xn}\left\{ x_{n}\right\} is a basis for XX, then {SN}\left\{ S_{N}\right\} is a family of partial sum projections.

(b)

Suppose that {PN}\left\{ P_{N}\right\} is a family of partial sum projections. Show that if there exist nonzero vectors x1∈range⁡(P1)x_{1} \in \operatorname {range}\left(P_{1}\right) and xn∈range⁡(Pn)∩ker⁡(Pn−1)x_{n} \in \operatorname {range}\left(P_{n}\right) \cap \operatorname {ker}\left(P_{n-1}\right) for n>1n > 1, then {xn}\left\{ x_{n}\right\} is a basis for XX.

Problem 5.7

Prove the remaining implications in Theorem 5.12.

?
Remark.
?

Theorem 5.12: Given a sequence {xn}\left\{ x_{n}\right\} in a Banach space XX, the following statements are equivalent.

  1. {xn}\left\{ x_{n}\right\} is a basis for XX.

  2. There exists a biorthogonal sequence {an}⊆X∗\left\{ a_{n}\right\} \subseteq X^{*} such that x=lim⁡N→∞SNx=∑n=1∞⟨x,an⟩xnx = \lim_{N \rightarrow \infty } S_{N} x = \sum_{n = 1}^{\infty } \left\langle x, a_{n} \right\rangle x_{n} for all x∈Xx \in X.

  3. {xn}\left\{ x_{n}\right\} is complete and there exists a biorthogonal sequence {an}⊆X∗\left\{ a_{n}\right\} \subseteq X^{*} such that the series ∑⟨x,an⟩xn\sum \left\langle x, a_{n} \right\rangle x_{n} converges for each x∈Xx \in X.

  4. {xn}\left\{ x_{n}\right\} is exact and sup⁡N∥SNx∥<∞\sup_{N}\left\| S_{N} x\right\| < \infty for all x∈Xx \in X.

  5. {xn}\left\{ x_{n}\right\} is exact and sup⁡N∥SN∥<∞\sup_{N}\left\| S_{N}\right\| < \infty. Here, for a minimal sequence {xn}\left\{ x_{n}\right\} with biorthogonal sequence {an}\left\{ a_{n}\right\}, SNx=∑n=1N⟨x,an⟩xnS_{N} x = \sum_{n = 1}^{N} \left\langle x, a_{n} \right\rangle x_{n}, x∈Xx \in X, are the partial sum operators. The text proves the implication (e) ⇒\Rightarrow (b).

Problem 5.8

Let {xn}\left\{ x_{n}\right\} be a basis for a Hilbert space HH and {yn}\left\{ y_{n}\right\} be a basis for a Hilbert space KK. This exercise will show that the tensor product sequence {xm⊗yn}m,n∈N\left\{ x_{m} \otimes y_{n}\right\}_{m, n \in \mathbb {N}} is a basis for the tensor product space H⊗K=B2(H,K)H \otimes K = \mathcal{B}_{2}(H, K) (see Appendix B for definitions).

Let {an},{bn}\left\{ a_{n}\right\} ,\left\{ b_{n}\right\} and CX,CY\mathcal{C}_{X}, \mathcal{C}_{Y} denote the biorthogonal systems and basis constants for {xn},{yn}\left\{ x_{n}\right\} ,\left\{ y_{n}\right\}, respectively. Order N×N\mathbb {N} \times \mathbb {N} as follows:

\begin{aligned} \mathbb {N} \times \mathbb {N} = & \left\{ (1,1), \\ & (2,1),(2,2),(1,2), \\ & (3,1),(3,2),(3,3),(2,3),(1,3), \\ & (4,1),(4,2),(4,3),(4,4),(3,4),(2,4),(1,4), \\ & \quad \cdots \right\} , \end{aligned}

and let {zk}k∈N\left\{ z_{k}\right\}_{k \in \mathbb {N}} denote {xm⊗yn}m,n∈N\left\{ x_{m} \otimes y_{n}\right\}_{m, n \in \mathbb {N}} arranged according to the above ordering of N×N\mathbb {N} \times \mathbb {N}. Let SNX,SNYS_{N}^{X}, S_{N}^{Y}, and SNZS_{N}^{Z} denote the partial sum operators associated with {xm},{yn}\left\{ x_{m}\right\} ,\left\{ y_{n}\right\}, and {zk}\left\{ z_{k}\right\}, respectively.

?
(a)

Show that {xm⊗yn}m,n∈N\left\{ x_{m} \otimes y_{n}\right\}_{m, n \in \mathbb {N}} is exact in H⊗KH \otimes K, and its biorthogonal system is {am⊗bn}m,n∈N\left\{ a_{m} \otimes b_{n}\right\}_{m, n \in \mathbb {N}}.

(b)

Prove the following relationships among the partial sum operators (where we let S0XS_{0}^{X} and S0YS_{0}^{Y} be the zero operators):

SN2Z=SNX⊗SNY,SN2+ℓZ=SNX⊗SNY+(SN+1X−SNX)⊗SℓY,ℓ=1,…,N+1,SN2+N+1+ℓZ=SN+1X⊗SN+1Y−SℓX⊗(SN+1Y−SNY),ℓ=1,…,N. \begin{aligned} S_{N^{2}}^{Z} & = S_{N}^{X} \otimes S_{N}^{Y}, \\ S_{N^{2}+\ell }^{Z} & = S_{N}^{X} \otimes S_{N}^{Y}+\left(S_{N+1}^{X}-S_{N}^{X}\right) \otimes S_{\ell }^{Y}, \quad \ell = 1, \ldots , N+1, \\ S_{N^{2}+N+1+\ell }^{Z} & = S_{N+1}^{X} \otimes S_{N+1}^{Y}-S_{\ell }^{X} \otimes \left(S_{N+1}^{Y}-S_{N}^{Y}\right), \quad \ell = 1, \ldots , N. \end{aligned}
(c)

Show that ∥SNZ∥≤3CXCY\left\| S_{N}^{Z}\right\| \leq 3 \mathcal{C}_{X} \mathcal{C}_{Y} for each N∈NN \in \mathbb {N}, and conclude that {zk}\left\{ z_{k}\right\} is a basis for H⊗KH \otimes K.

Problem 5.9

Fix 0<α<1/20 < \alpha < 1 / 2, and let φ(t)=∣t−12∣α\varphi (t) = \left|t-\frac{1}{2}\right|^{\alpha } be as in Example 5.13.

?
(a)

Prove that {e2πintφ(t)}n∈Z\left\{ e^{2 \pi i n t} \varphi (t)\right\}_{n \in \mathbb {Z}} is exact in L2(T)L^{2}(\mathbf{T}), and its biorthogonal system {e2πintφ~(t)}n∈Z\left\{ e^{2 \pi i n t} \widetilde{\varphi }(t)\right\}_{n \in \mathbb {Z}} is also exact (compare Exercise 10.9).

(b)

Prove that ∣φ∣2∈A2(T)\left|\varphi \right|^{2} \in \mathcal{A}_{2}(\mathbf{T}).

Problem 5.10

Let XX be a complex Banach space. By Exercise 1.7, the vector space XR=XX_{\mathbb {R}} = X over the real field is a real Banach space. Let {xn}\left\{ x_{n}\right\} be a fixed sequence in XX. Show that

{xn} is a basis for X⟺{x1,ix1,x2,ix2,…} is a basis for XR. \left\{ x_{n}\right\} \text{ is a basis for } X \Longleftrightarrow \left\{ x_{1}, i x_{1}, x_{2}, i x_{2}, \ldots \right\} \text{ is a basis for } X_{\mathbb {R}}.
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Problem 5.11

Let XX be a Banach space. Show that {xn}⊆X\left\{ x_{n}\right\} \subseteq X is a monotone basis for XX if and only if {xn}\left\{ x_{n}\right\} is complete, xn≠0x_{n} \neq 0 for every nn, and

∀N∈N,∀c1,…,cN,cN+1∈F,∥∑n=1Ncnxn∥≤∥∑n=1N+1cnxn∥. \forall N \in \mathbb {N}, \quad \forall c_{1}, \ldots , c_{N}, c_{N+1} \in \mathbf{F}, \quad \left\| \sum _{n = 1}^{N} c_{n} x_{n}\right\| \leq \left\| \sum _{n = 1}^{N+1} c_{n} x_{n}\right\| .
?
Problem 5.12

Prove that the Schauder system is a monotone basis for C[0,1]C[0,1].

?
Problem 5.13

Let XX be a Banach space. Show that XX has a monotone basis if and only if there exists a sequence of operators {PN}⊆B(X)\left\{ P_{N}\right\} \subseteq \mathcal{B}(X) such that for each N∈NN \in \mathbb {N} we have

  1. ∥PN∥=1\left\| P_{N}\right\| = 1,

  2. PN2=PNP_{N}^{2} = P_{N},

  3. dim⁡(range⁡(PN))=N\operatorname {dim}\left(\operatorname {range}\left(P_{N}\right)\right) = N,

  4. range⁡(PN)⊆range⁡(PN+1)\operatorname {range}\left(P_{N}\right) \subseteq \operatorname {range}\left(P_{N+1}\right), and

  5. ⋃range⁡(PN)\bigcup \operatorname {range}\left(P_{N}\right) is dense in XX.

?
Problem 5.14

Fix 1≤p<∞1 \leq p < \infty, and prove the following statements.

?
(a)

(1+t)p≤2p−1(1+tp)(1+t)^{p} \leq 2^{p-1}\left(1+t^{p}\right) for all t≥1t \geq 1.

(b)

If a,b∈Fa, b \in \mathbf{F} then ∣a+b∣p≤2p−1(∣a∣p+∣b∣p)\left|a+b\right|^{p} \leq 2^{p-1}\left(\left|a\right|^{p}+\left|b\right|^{p}\right).

(c)

The quantity appearing in equation (5.10) is nonnegative.

Problem 5.15

Let {hn}\left\{ h_{n}\right\} be the Haar system, enumerated as in equation (5.9).

?
(a)

What is the biorthogonal system for {hn}\left\{ h_{n}\right\} as a basis for Lp[0,1]L^{p}[0,1]?

(b)

Show that {hn}\left\{ h_{n}\right\} is a basic sequence in L∞[0,1]L^{\infty }[0,1].

(c)

Show that C[0,1]⊆span⁡‾{hn}C[0,1] \subseteq \overline{\operatorname {span}}\left\{ h_{n}\right\} (closure in L∞L^{\infty }-norm). In fact, show that if f∈C[0,1]f \in C[0,1] then the series ∑⟨f,hn⟩hn\sum \left\langle f, h_{n} \right\rangle h_{n} converges uniformly to ff.

Problem 5.16
?
(a)

Suppose that f∈C(R)f \in C(\mathbb {R}) is Hölder continuous with exponent α\alpha, i.e., there exists a constant K>0K > 0 such that ∣f(x)−f(y)∣≤K∣x−y∣α\left|f(x)-f(y)\right| \leq K\left|x-y\right|^{\alpha } for all xx, y∈[0,1]y \in [0,1] (compare Exercise 1.23). Show that there exists a constant C>0C > 0 such that

∣⟨f,ψn,k⟩∣≤C2−n(α+1/2),n≥0,k∈Z \left| \left\langle f, \psi _{n, k} \right\rangle \right| \leq C 2^{-n(\alpha +1 / 2)}, \quad n \geq 0, k \in \mathbb {Z}
(b)

How does the conclusion change if we only assume that ff is Hölder continuous at a point xx, i.e., there exist K,δ>0K, \delta > 0 such that ∣f(x)−f(y)∣≤K∣x−y∣α\left|f(x)-f(y)\right| \leq K\left|x-y\right|^{\alpha } for all yy with ∣x−y∣<δ\left|x-y\right| < \delta?

Problem 5.17

Let {xn}\left\{ x_{n}\right\} be a basis for a Banach space XX, with biorthogonal system {an}⊆X∗\left\{ a_{n}\right\} \subseteq X^{*}.

?
(a)

Show that if {xn}\left\{ x_{n}\right\} is a bounded basis for XX, then {an}\left\{ a_{n}\right\} is a bounded basis for span⁡‾{an}\overline{\operatorname {span}}\left\{ a_{n}\right\} in X∗X^{*}.

(b)

If {xn}\left\{ x_{n}\right\} is a normalized basis for XX, must {an}\left\{ a_{n}\right\} be a normalized basis for span⁡‾{an}\overline{\operatorname {span}}\left\{ a_{n}\right\}?

Problem 5.18

Suppose that {xn}\left\{ x_{n}\right\} is a minimal system in a Banach space XX, and it has a biorthogonal system {an}\left\{ a_{n}\right\} that is a basis for X∗X^{*}. Prove that {xn}\left\{ x_{n}\right\} is a basis for XX.

?
Problem 5.19
?
(a)

Prove Corollary 5.25.

(b)

Does Theorem 5.24 or Corollary 5.25 remain valid if λ=1\lambda = 1?

Problem 5.20
?
(a)

Prove the Krein-Milman-Rutman Theorem: If {xn}\left\{ x_{n}\right\} is a basis for a Banach space XX, then there exist constants εn\varepsilon_{n} such that if {yn}\left\{ y_{n}\right\} is a sequence in XX satisfying ∥xn−yn∥<εn\left\| x_{n}-y_{n}\right\| < \varepsilon_{n} for all nn, then {yn}\left\{ y_{n}\right\} is a basis for XX that is equivalent to {xn}\left\{ x_{n}\right\}.

(b)

Suppose that XX is a Banach space that has a basis. Show that any dense subset EE of XX contains a basis for XX.

(c)

Show that there exists a basis {pn}\left\{ p_{n}\right\} for C[0,1]C[0,1] such that each pnp_{n} is a polynomial. Contrast this with the fact that {xk}k≥0\left\{ x^{k}\right\}_{k \geq 0} is not a basis for C[0,1]C[0,1], and the Schauder system is a basis for C[0,1]C[0,1] that does not consist of polynomials.

Problem 5.21

This exercise defines and presents some facts about quotient spaces for vector spaces analogous to those for quotient groups in abstract algebra.

Let MM be a subspace of a vector space VV.

?
(a)

Define x∼yx \sim y if x−y∈Mx-y \in M. Show that ∼\sim is an equivalence relation on VV, and the equivalence class of xx is [x]=M+x={m+x:m∈M}[x] = M+x = \left\{ m+x: m \in M\right\}.

(b)

The quotient space is V/M={M+x:x∈V}V / M = \left\{ M+x: x \in V\right\}. Show that

(x+M)+(y+M)=(x+y)+M and c(x+M)=(cx)+M (x+M)+(y+M) = (x+y)+M \quad \text{ and } \quad c(x+M) = (c x)+M

are well-defined operations on V/MV / M, and V/MV / M is a vector space with respect to these operations.

(c)

Prove the Isomorphism Theorem for vector spaces: If V,XV, X are vector spaces and T:V→WT: V \rightarrow W is linear and surjective with kernel⁡M=ker⁡(T)\operatorname {kernel} M = \operatorname {ker}(T), then ψ(x+M)=Tx\psi (x+M) = T x is a well-defined linear bijection of V/MV / M onto WW.

(d)

If there exists a subspace N⊆VN \subseteq V such that M∩N={0}M \cap N = \left\{ 0\right\} and M+N={m+n:m∈M,n∈N}=VM+N = \left\{ m+n: m \in M, n \in N\right\} = V, then we define the codimension of MM to be codim⁡(M)=dim⁡(N)\operatorname {codim}(M) = \operatorname {dim}(N). Show that the codimension is independent of the choice of subspace NN, and codim⁡(M)=dim⁡(V/M)\operatorname {codim}(M) = \operatorname {dim}(V / M).

Chapter 6Unconditional Bases in Banach Spaces[8/100%]
Problem 6.1

Prove Lemma 6.1.

?
Remark.
?

Lemma 6.1: Given a sequence {xn}\left\{ x_{n}\right\} in a Banach space XX, the following two statements are equivalent: (a) {xn}\left\{ x_{n}\right\} is an unconditional basis for XX; (b) {xσ(n)}\left\{ x_{\sigma (n)}\right\} is a basis for XX for every permutation σ\sigma of N\mathbb {N}. In this case, if {an}\left\{ a_{n}\right\} is the sequence of coefficient functionals for {xn}\left\{ x_{n}\right\}, then {aσ(n)}\left\{ a_{\sigma (n)}\right\} is the sequence of coefficient functionals for {xσ(n)}\left\{ x_{\sigma (n)}\right\}.

Problem 6.2

Prove Lemma 6.2: let T:X→YT: X \rightarrow Y be a topological isomorphism of Banach spaces and let {xn}\left\{ x_{n}\right\} be a basis for XX.

?
(a)

If {xn}\left\{ x_{n}\right\} is an unconditional basis for XX, then {Txn}\left\{ T x_{n}\right\} is an unconditional basis for YY.

(b)

If {xn}\left\{ x_{n}\right\} is a bounded unconditional basis for XX, then {Txn}\left\{ T x_{n}\right\} is a bounded unconditional basis for YY.

Problem 6.3

Prove Theorem 6.4.

?
Remark.
?

Notation 6.3: Let {xn}\left\{ x_{n}\right\} be an unconditional basis for a Banach space XX, and let {an}\left\{ a_{n}\right\} be the biorthogonal system to {xn}\left\{ x_{n}\right\}. To each finite set F⊆NF \subseteq \mathbb {N} we associate the partial sum operator SF(x)=∑n∈F⟨x,an⟩xnS_{F}(x) = \sum_{n \in F} \left\langle x, a_{n} \right\rangle x_{n}; to each finite set F⊆NF \subseteq \mathbb {N} and each set of scalars E={εn}n∈F\mathcal{E} = \left\{ \varepsilon_{n}\right\}_{n \in F} satisfying εn=±1\varepsilon_{n} = \pm 1 for all nn, the operator SF,E(x)=∑n∈Fεn⟨x,an⟩xnS_{F, \mathcal{E}}(x) = \sum_{n \in F} \varepsilon_{n} \left\langle x, a_{n} \right\rangle x_{n}; and to each finite set F⊆NF \subseteq \mathbb {N} and each collection of scalars Λ={λn}n∈F\Lambda = \left\{ \lambda_{n}\right\}_{n \in F} satisfying ∣λn∣≤1\left|\lambda_{n}\right| \leq 1 for all nn, the operator SF,Λ(x)=∑n∈Fλn⟨x,an⟩xnS_{F, \Lambda }(x) = \sum_{n \in F} \lambda_{n} \left\langle x, a_{n} \right\rangle x_{n} (all for x∈Xx \in X). The suprema below are taken over all such FF, E\mathcal{E}, Λ\Lambda.

Theorem 6.4: If {xn}\left\{ x_{n}\right\} is an unconditional basis for a Banach space XX, then the following statements hold.

  1. The following three quantities are finite for each x∈Xx \in X:

    ∣∣∣x∣∣∣=sup⁡F∥SF(x)∥,∣∣∣x∣∣∣E=sup⁡F,E∥SF,E(x)∥,∣∣∣x∣∣∣Λ=sup⁡F,Λ∥SF,Λ(x)∥. |||x||| = \sup _{F}\left\| S_{F}(x)\right\| , \quad |||x|||_{\mathcal{E}} = \sup _{F, \mathcal{E}}\left\| S_{F, \mathcal{E}}(x)\right\| , \quad |||x|||_{\Lambda } = \sup _{F, \Lambda }\left\| S_{F, \Lambda }(x)\right\| .

  2. The following three numbers are finite:

    K=sup⁡F∥SF∥,KE=sup⁡F,E∥SF,E∥,KΛ=sup⁡F,Λ∥SF,Λ∥. \mathcal{K} = \sup _{F}\left\| S_{F}\right\| , \quad \mathcal{K}_{\mathcal{E}} = \sup _{F, \mathcal{E}}\left\| S_{F, \mathcal{E}}\right\| , \quad \mathcal{K}_{\Lambda } = \sup _{F, \Lambda }\left\| S_{F, \Lambda }\right\| .

  3. ∣∣∣⋅∣∣∣≤∣∣∣⋅∣∣∣E≤2 ∣∣∣⋅∣∣∣|||\cdot ||| \leq |||\cdot |||_{\mathcal{E}} \leq 2\, |||\cdot ||| and K≤KE≤2K\mathcal{K} \leq \mathcal{K}_{\mathcal{E}} \leq 2 \mathcal{K}.

  4. If F=R\mathbf{F} = \mathbb {R} then ∣∣∣⋅∣∣∣E=∣∣∣⋅∣∣∣Λ|||\cdot |||_{\mathcal{E}} = |||\cdot |||_{\Lambda } and KE=KΛ\mathcal{K}_{\mathcal{E}} = \mathcal{K}_{\Lambda }.

  5. If F=C\mathbf{F} = \mathbb {C} then ∣∣∣⋅∣∣∣E≤∣∣∣⋅∣∣∣Λ≤2 ∣∣∣⋅∣∣∣E|||\cdot |||_{\mathcal{E}} \leq |||\cdot |||_{\Lambda } \leq 2\, |||\cdot |||_{\mathcal{E}} and KE≤KΛ≤2KE\mathcal{K}_{\mathcal{E}} \leq \mathcal{K}_{\Lambda } \leq 2 \mathcal{K}_{\mathcal{E}}.

  6. ∣∣∣⋅∣∣∣|||\cdot |||, ∣∣∣⋅∣∣∣E|||\cdot |||_{\mathcal{E}}, and ∣∣∣⋅∣∣∣Λ|||\cdot |||_{\Lambda } form norms on XX, each equivalent to the initial norm ∥⋅∥\left\| \cdot \right\|, with

    ∥⋅∥≤∣∣∣⋅∣∣∣≤K∥⋅∥,∥⋅∥≤∣∣∣⋅∣∣∣E≤KE∥⋅∥,∥⋅∥≤∣∣∣⋅∣∣∣Λ≤KΛ∥⋅∥. \left\| \cdot \right\| \leq |||\cdot ||| \leq \mathcal{K}\left\| \cdot \right\| , \quad \left\| \cdot \right\| \leq |||\cdot |||_{\mathcal{E}} \leq \mathcal{K}_{\mathcal{E}}\left\| \cdot \right\| , \quad \left\| \cdot \right\| \leq |||\cdot |||_{\Lambda } \leq \mathcal{K}_{\Lambda }\left\| \cdot \right\| .
Problem 6.4

Let {xn}\left\{ x_{n}\right\} be an unconditional basis for a Banach space XX, with associated coefficient functionals {an}\left\{ a_{n}\right\}.

?
(a)

Prove that {an}\left\{ a_{n}\right\} is an unconditional basic sequence in X∗X^{*}.

(b)

Show that if XX is reflexive, then {an}\left\{ a_{n}\right\} is an unconditional basis for X∗X^{*}.

Problem 6.5

(See Chapter 14 for proof that {e2πint}n∈Z\left\{ e^{2 \pi i n t}\right\}_{n \in \mathbb {Z}} is a basis for Lp(T)L^{p}(\mathbf{T}) when 1<p<∞1 < p < \infty, but is not a basis for L1(T)L^{1}(\mathbf{T}) or C(T)C(\mathbf{T}).)

?
(a)

Use Orlicz's Theorem to prove that {e2πint}n∈Z\left\{ e^{2 \pi i n t}\right\}_{n \in \mathbb {Z}} cannot be an unconditional basis for Lp(T)L^{p}(\mathbf{T}) when 1≤p<21 \leq p < 2.

(b)

Argue by duality to show that it also cannot be an unconditional basis when 2<p<∞2 < p < \infty.

Problem 6.6

Let XX be a real Banach space, and suppose that {xn}\left\{ x_{n}\right\} is an unconditional basis for XX with unconditional basis constant KE=1\mathcal{K}_{\mathcal{E}} = 1. Given x=∑anxnx = \sum a_{n} x_{n} and y=∑bnyny = \sum b_{n} y_{n} in XX, declare that x≤yx \leq y if an≤bna_{n} \leq b_{n} for every nn. Show that ≤\leq is a partial order on XX, and XX is a Banach lattice in the sense of Definition 3.35. Using the notation of that definition, show that x∨y=∑max⁡{an,bn}xnx \vee y = \sum \max \left\{ a_{n}, b_{n}\right\} x_{n}, x∧y=∑min⁡{an,bn}xnx \wedge y = \sum \min \left\{ a_{n}, b_{n}\right\} x_{n}, and ∣x∣=∑∣an∣xn\left|x\right| = \sum \left|a_{n}\right| x_{n}.

?
Problem 6.7

Set F=RF = \mathbb {R}. The Haar system is an orthonormal basis for L2[0,1]L^{2}[0,1], so by Exercise 6.6 there is a partial ordering ≤\leq on L2[0,1]L^{2}[0,1] induced by this unconditional basis. There is also the ordinary partial ordering ≤\leq on L2[0,1]L^{2}[0,1] defined by f≤gf \leq g if f(t)≤g(t)f(t) \leq g(t) for a.e. tt. Do these two orderings coincide?

?
Problem 6.8

Let WW be the hat function of height 1 supported on [0,1][0,1], and let sn,k(t)=W(2nt−k)s_{n, k}(t) = W\left(2^{n} t-k\right) be the dilated and translated hat functions of the Schauder system in C[0,1]C[0,1]. Select the subsequence

t1=s0,0,t2=s1,0,t3=s2,1,t4=s3,2,t5=s4,5,t6=s5,10,…, t_{1} = s_{0,0}, \quad t_{2} = s_{1,0}, \quad t_{3} = s_{2,1}, \quad t_{4} = s_{3,2}, \quad t_{5} = s_{4,5}, \quad t_{6} = s_{5,10}, \quad \ldots ,

the hat functions on the intervals I1=[0,1]I_{1} = [0,1], I2=[0,12]I_{2} = \left[0, \frac{1}{2}\right], I3=[14,12]I_{3} = \left[\frac{1}{4}, \frac{1}{2}\right], I4=[14,38]I_{4} = \left[\frac{1}{4}, \frac{3}{8}\right], I5=[516,38]I_{5} = \left[\frac{5}{16}, \frac{3}{8}\right], I6=[516,1132]I_{6} = \left[\frac{5}{16}, \frac{11}{32}\right], etc., where we alternate choosing the left or right half of IN−1I_{N-1} as the interval INI_{N} on which the hat function tNt_{N} is supported. Let

gN=∑n=1Ntn,hN=∑n=1N(−1)n+1tn, g_{N} = \sum _{n = 1}^{N} t_{n}, \quad h_{N} = \sum _{n = 1}^{N}(-1)^{n+1} t_{n},

let μN\mu_{N} be the midpoint of INI_{N}, let aN=gN(μN)a_{N} = g_{N}\left(\mu_{N}\right) be the global maximum of gNg_{N}, and let bN=∥hN∥∞b_{N} = \left\| h_{N}\right\|_{\infty }. (For N≥3N \geq 3 one endpoint of INI_{N} is μN−2\mu_{N-2} and the other is μN−1\mu_{N-1}, and aN=1+aN−1+aN−22a_{N} = 1+\frac{a_{N-1}+a_{N-2}}{2}.)

?
(a)

Show that aN→∞a_{N} \rightarrow \infty.

(b)

Show that 0≤bN≤20 \leq b_{N} \leq 2 for each NN.

Chapter 7Bessel Sequences and Bases in Hilbert Spaces[20/100%]
Problem 7.1

Let HH be a separable Hilbert space. For each of the following, construct a sequence {xn}\left\{ x_{n}\right\} that has the specified property.

?
(a)

A bounded sequence that is not a Bessel sequence.

(b)

A Bessel sequence that is a nonorthogonal basis for HH.

(c)

A Bessel sequence that is not a basis for HH but has frame operator S=IS = I.

(d)

A Bessel sequence such that {xn}n∈N\F\left\{ x_{n}\right\}_{n \in \mathbb {N} \backslash F} is complete for every finite F⊆ NF \subseteq \mathrm{~ N}.

(e)

An unconditional basis that is not a Bessel sequence.

(f)

A normalized conditional basis that is a Bessel sequence.

(g)

A normalized conditional basis that is not a Bessel sequence.

(h)

A basis that is Bessel but whose biorthogonal sequence is not Bessel.

Problem 7.2

Give a direct proof of Theorem 7.2.

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Remark.
?

Theorem 7.2: Let {xn}\left\{ x_{n}\right\} be a Bessel sequence in a Hilbert space HH. If we define Cx=⟨x,xn⟩C x = \left\langle x, x_{n} \right\rangle for x∈Hx \in H, then the following statements hold. (a) CC is a bounded mapping of HH into ℓ2\ell^{2}, and therefore there exists a constant B>0B > 0 such that ∑n∣⟨x,xn⟩∣2≤B∥x∥2\sum_{n}\left| \left\langle x, x_{n} \right\rangle \right|^{2} \leq B\left\| x\right\|^{2} for all x∈Hx \in H. (b) If (cn)∈ℓ2(c_{n}) \in \ell^{2}, then the series ∑cnxn\sum c_{n} x_{n} converges unconditionally in HH, and Rc=∑cnxnR c = \sum c_{n} x_{n} defines a bounded map of ℓ2\ell^{2} into HH. (c) R=C∗R = C^{*} and ∥R∥=∥C∥≤B1/2\left\| R\right\| = \left\| C\right\| \leq B^{1 / 2}. (d) If {xn}\left\{ x_{n}\right\} is complete, then CC is injective and range⁡(R)\operatorname {range}(R) is dense in HH.

Problem 7.3

Prove Theorem 7.4.

?
Remark.
?

Theorem 7.4: Let {xn}\left\{ x_{n}\right\} be a sequence in a Hilbert space HH, and let {en}\left\{ e_{n}\right\} be the standard basis for ℓ2\ell^{2}. Then the following statements are equivalent. (a) {xn}\left\{ x_{n}\right\} is a Bessel sequence in HH. (b) There exists a constant B>0B > 0 and a dense set E⊆HE \subseteq H such that ∑n∣⟨x,xn⟩∣2≤B∥x∥2\sum_{n}\left| \left\langle x, x_{n} \right\rangle \right|^{2} \leq B\left\| x\right\|^{2} for all x∈Ex \in E. (c) There exists a constant B>0B > 0 such that ∥∑n=1Ncnxn∥2≤B∑n=1N∣cn∣2\left\| \sum_{n = 1}^{N} c_{n} x_{n}\right\|^{2} \leq B \sum_{n = 1}^{N}\left|c_{n}\right|^{2} for every N∈NN \in \mathbb {N} and c1,…,cN∈Fc_{1}, \ldots , c_{N} \in \mathbf{F}. (d) The series ∑cnxn\sum c_{n} x_{n} converges for each sequence (cn)∈ℓ2(c_{n}) \in \ell^{2}. (e) There exists a bounded operator R:ℓ2→HR: \ell^{2} \rightarrow H such that Ren=xnR e_{n} = x_{n} for each n∈Nn \in \mathbb {N}. Further, when these hold, the operator RR appearing in part (e) is the synthesis operator for {xn}\left\{ x_{n}\right\}.

Problem 7.4

Prove Theorem 7.5.

?
Remark.
?

Theorem 7.5: Let {xn}\left\{ x_{n}\right\} be a Bessel sequence in a Hilbert space HH. Then the matrix for the Gram operator GG is G=(⟨xn,xm⟩)m,n∈NG = \left( \left\langle x_{n}, x_{m} \right\rangle \right)_{m, n \in \mathbb {N}}.

Problem 7.5

Let {xn}\left\{ x_{n}\right\} be a Bessel sequence in a Hilbert space HH and let BB be a Bessel bound.

?
(a)

Show that ∥xn∥2≤B\left\| x_{n}\right\|^{2} \leq B for every n∈Nn \in \mathbb {N}. Thus Bessel sequences are bounded above in norm.

(b)

Show that if ∥xm∥2=B\left\| x_{m}\right\|^{2} = B for any particular mm, then xn⊥xmx_{n} \perp x_{m} for all n≠mn \neq m.

Problem 7.6

Let H,KH, K be Hilbert spaces. Show that if {xn}\left\{ x_{n}\right\} is a Bessel sequence in HH and L∈B(H,K)L \in \mathcal{B}(H, K), then {Lxn}\left\{ L x_{n}\right\} is a Bessel sequence in KK.

?
Problem 7.7

Suppose that HH is a Hilbert space contained in another Hilbert space KK. Given a sequence {xn}\left\{ x_{n}\right\} in HH, show that {xn}\left\{ x_{n}\right\} is a Bessel sequence in HH if and only if it is a Bessel sequence in KK.

?
Problem 7.8

Let {xn}\left\{ x_{n}\right\} be a sequence in a Hilbert space HH.

?
(a)

If ∑∣⟨x,xn⟩∣2<∞\sum \left| \left\langle x, x_{n} \right\rangle \right|^{2} < \infty for all xx in a dense set E⊆HE \subseteq H, must {xn}\left\{ x_{n}\right\} be a Bessel sequence?

(b)

If there exists a constant B>0B > 0 such that ∑∣⟨x,xn⟩∣2≤B∥x∥2\sum \left| \left\langle x, x_{n} \right\rangle \right|^{2} \leq B\left\| x\right\|^{2} for all xx in a complete set E⊆HE \subseteq H, must {xn}\left\{ x_{n}\right\} be a Bessel sequence?

Problem 7.9

Show that a sequence {xn}\left\{ x_{n}\right\} in a Hilbert space HH is a Bessel sequence if either of the following two conditions holds:

?
(a)

∑m∑n∣⟨xm,xn⟩∣2<∞\sum_{m} \sum_{n}\left| \left\langle x_{m}, x_{n} \right\rangle \right|^{2} < \infty, or

(b)

sup⁡m∑n∣⟨xm,xn⟩∣<∞\sup_{m} \sum_{n}\left| \left\langle x_{m}, x_{n} \right\rangle \right| < \infty.

Observe that hypothesis (a) is quite restrictive, e.g., it is not satisfied by any infinite orthonormal sequence.

Problem 7.10

Suppose that {xn}\left\{ x_{n}\right\} is a Bessel sequence that is a basis for a Hilbert space HH. Let {yn}\left\{ y_{n}\right\} be the biorthogonal sequence, and let BB be a Bessel bound.

?
(a)

Show that

∀x∈H,1B∥x∥2≤∑n∣⟨x,yn⟩∣2 \forall x \in H, \quad \frac{1}{B}\left\| x\right\| ^{2} \leq \sum _{n}\left| \left\langle x, y_{n} \right\rangle \right|^{2}

We say that {yn}\left\{ y_{n}\right\} has a lower frame bound of B−1B^{-1}; compare Definition 8.2. Note that {yn}\left\{ y_{n}\right\} need not be a Bessel sequence; see Exercise 7.1(h).

(b)

Show that for all N∈NN \in \mathbb {N} and c1,…,cN∈Fc_{1}, \ldots , c_{N} \in \mathbf{F} we have

1B∑n=1N∣cn∣2≤∥∑n=1Ncnyn∥2 \frac{1}{B} \sum _{n = 1}^{N}\left|c_{n}\right|^{2} \leq \left\| \sum _{n = 1}^{N} c_{n} y_{n}\right\| ^{2}
Problem 7.11

Let {xn},{yn}\left\{ x_{n}\right\} ,\left\{ y_{n}\right\} be Bessel sequences in separable Hilbert spaces H,KH, K, respectively. Show that the tensor product sequence {xm⊗yn}m,n∈N\left\{ x_{m} \otimes y_{n}\right\}_{m, n \in \mathbb {N}} is a Bessel sequence in H⊗K=B2(H,K)H \otimes K = \mathcal{B}_{2}(H, K) (see Appendix B for definitions).

?
Problem 7.12

The Hilbert matrix is

H=[11/21/31/4⋯1/21/31/41/51/31/41/51/61/41/51/61/7⋮⋱] \mathcal{H} = \left[\begin{smallmatrix} 1 & 1 / 2 & 1 / 3 & 1 / 4 & \cdots \\ 1 / 2 & 1 / 3 & 1 / 4 & 1 / 5 & \\ 1 / 3 & 1 / 4 & 1 / 5 & 1 / 6 & \\ 1 / 4 & 1 / 5 & 1 / 6 & 1 / 7 & \\ \vdots & & & & \ddots \end{smallmatrix}\right]

Define

C=[1000⋯1/21/2001/31/31/301/41/41/41/4⋮⋱] and L=[11/21/31/4⋯1/21/21/31/41/31/31/31/41/41/41/41/4⋮⋱] C = \left[\begin{smallmatrix} 1 & 0 & 0 & 0 & \cdots \\ 1 / 2 & 1 / 2 & 0 & 0 & \\ 1 / 3 & 1 / 3 & 1 / 3 & 0 & \\ 1 / 4 & 1 / 4 & 1 / 4 & 1 / 4 & \\ \vdots & & & & \ddots \end{smallmatrix}\right] \quad \text{ and } \quad L = \left[\begin{smallmatrix} 1 & 1 / 2 & 1 / 3 & 1 / 4 & \cdots \\ 1 / 2 & 1 / 2 & 1 / 3 & 1 / 4 & \\ 1 / 3 & 1 / 3 & 1 / 3 & 1 / 4 & \\ 1 / 4 & 1 / 4 & 1 / 4 & 1 / 4 & \\ \vdots & & & & \ddots \end{smallmatrix}\right]

and prove the following statements.

?
(a)

L=CC∗L = C C^{*}, so L≥0L \geq 0 (i.e., LL is a positive operator).

(b)

I−(I−C)(I−C)∗=diag⁡(1,1/2,1/3,1/4,…)I-(I-C)(I-C)^{*} = \operatorname {diag}(1,1 / 2,1 / 3,1 / 4, \ldots ), the diagonal matrix with entries 1,1/2,…1,1 / 2, \ldots on the diagonal.

(c)

∥(I−C)∥2=∥(I−C)(I−C)∗∥≤1\left\| (I-C)\right\|^{2} = \left\| (I-C)(I-C)^{*}\right\| \leq 1.

(d)

∥C∥≤2\left\| C\right\| \leq 2 and ∥L∥≤4\left\| L\right\| \leq 4.

Remark It is a fact (though not so easy to prove) that if A,BA, B are symmetric matrices and aij≤bija_{i j} \leq b_{i j} for all i,j∈Ni, j \in \mathbb {N}, then ∥A∥≤∥B∥\left\| A\right\| \leq \left\| B\right\|. Consequently, ∥H∥≤∥L∥≤4\left\| \mathcal{H}\right\| \leq \left\| L\right\| \leq 4.

Problem 7.13

Given a Riesz basis {xn}\left\{ x_{n}\right\} in a Hilbert space HH, prove that the following statements are equivalent.

?
(a)

∑cnxn\sum c_{n} x_{n} converges.

(b)

∑cnxn\sum c_{n} x_{n} converges unconditionally.

(c)

∑∣cn∣2<∞\sum \left|c_{n}\right|^{2} < \infty.

Problem 7.14

Show that every basis for a finite-dimensional vector space VV is a Riesz basis for VV (with respect to any inner product on VV).

?
Problem 7.15

Exhibit an unconditional basis for a Hilbert space HH that is not a Riesz basis for HH.

?
Problem 7.16

Show that if {xn}\left\{ x_{n}\right\} is a complete sequence in a Hilbert space HH that satisfies ∥∑n=1Ncnxn∥2=∑n=1N∣cn∣2\left\| \sum_{n = 1}^{N} c_{n} x_{n}\right\|^{2} = \sum_{n = 1}^{N}\left|c_{n}\right|^{2} for any N∈NN \in \mathbb {N} and c1,…,cN∈Fc_{1}, \ldots , c_{N} \in \mathbf{F}, then {xn}\left\{ x_{n}\right\} is an orthonormal basis for HH.

?
Problem 7.17

Let {xn},{yn}\left\{ x_{n}\right\} ,\left\{ y_{n}\right\} be Riesz bases for Hilbert spaces H,KH, K, respectively. Show that the tensor product sequence {xm⊗yn}m,n∈N\left\{ x_{m} \otimes y_{n}\right\}_{m, n \in \mathbb {N}} is a Riesz basis for H⊗K=B2(H,K)H \otimes K = \mathcal{B}_{2}(H, K) (see Appendix B for definitions).

?
Problem 7.18

Let {xn}\left\{ x_{n}\right\} be an orthonormal basis for a Hilbert space HH. Suppose {yn}\left\{ y_{n}\right\} is a sequence in HH and there exists 0<λ<10 < \lambda < 1 such that

∥∑n=1Ncn(xn−yn)∥2≤λ∑n=1N∣cn∣2,N∈N,c1,…,cN∈F \left\| \sum _{n = 1}^{N} c_{n}\left(x_{n}-y_{n}\right)\right\| ^{2} \leq \lambda \sum _{n = 1}^{N}\left|c_{n}\right|^{2}, \quad N \in \mathbb {N}, c_{1}, \ldots , c_{N} \in \mathbf{F}

Show that {yn}\left\{ y_{n}\right\} is a Riesz basis for HH.

?
Problem 7.19

Let {xn}\left\{ x_{n}\right\} be an orthonormal basis for a Hilbert space HH. Let Tk∈B(H)T_{k} \in \mathcal{B}(H) and ank∈Fa_{n k} \in \mathbf{F} be such that

λ=∑k=1∞∥Tk∥(sup⁡n∣ank∣)<1. \lambda = \sum _{k = 1}^{\infty }\left\| T_{k}\right\| \left(\sup _{n}\left|a_{n k}\right|\right) < 1.

Assume that the series

yn=xn+∑k=1∞ankTken y_{n} = x_{n}+\sum _{k = 1}^{\infty } a_{n k} T_{k} e_{n}

converges for each n∈Nn \in \mathbb {N}. Show that {yn}\left\{ y_{n}\right\} is a Riesz basis for HH.

?
Problem 7.20

In this exercise we will use the abbreviation eb(x)=e2πibxe_{b}(x) = e^{2 \pi i b x}, where b∈Rb \in \mathbb {R}. Also, we identify the Hilbert space L2(T)L^{2}(\mathbf{T}) with L2[−12,12]L^{2}\left[-\frac{1}{2}, \frac{1}{2}\right].

Fix λn∈C\lambda_{n} \in \mathbb {C} and assume that

δ=sup⁡n∈Z∣n−λn∣<∞ \delta = \sup _{n \in \mathbb {Z}}\left|n-\lambda _{n}\right| < \infty
?
(a)

Define bounded linear operators TkT_{k} on L2[−12,12]L^{2}\left[-\frac{1}{2}, \frac{1}{2}\right] by

Tkf(x)=xkf(x) T_{k} f(x) = x^{k} f(x)

Show that the operator norm of TkT_{k} is ∥Tk∥=2−k\left\| T_{k}\right\| = 2^{-k}.

(b)

Define

ank=−(2πi(λn−n))kk! a_{n k} = -\frac{\left(2 \pi i\left(\lambda _{n}-n\right)\right)^{k}}{k!}

Show that

en−eλn=∑k=1∞ankTken,n∈Z e_{n}-e_{\lambda _{n}} = \sum _{k = 1}^{\infty } a_{n k} T_{k} e_{n}, \quad n \in \mathbb {Z}

where the series converge absolutely in L2[−12,12]L^{2}\left[-\frac{1}{2}, \frac{1}{2}\right].

(c)

Show that if

δ<(ln⁡2)/π≈0.22… \delta < (\ln 2) / \pi \approx 0.22 \ldots

then {e2πiλnx}n∈Z\left\{ e^{2 \pi i \lambda_{n} x}\right\}_{n \in \mathbb {Z}} is a Riesz basis for L2[−12,12]L^{2}\left[-\frac{1}{2}, \frac{1}{2}\right].

Remark This result is due to Duffin and Eachus [DE42], but it is not quite the best possible. Kadec's 14\frac{1}{4}-Theorem [Kad64] states that if δ<14\delta < \frac{1}{4} then {e2πiλnx}n∈Z\left\{ e^{2 \pi i \lambda_{n} x}\right\}_{n \in \mathbb {Z}} is a Riesz basis for L2[−12,12]L^{2}\left[-\frac{1}{2}, \frac{1}{2}\right], and it is known that 14\frac{1}{4} is the optimal value. For a more detailed discussion, we refer to [You01].

Chapter 8Frames in Hilbert Spaces[42/100%]
Problem 8.1

Prove that the Mercedes frame is a tight frame for R2\mathbb {R}^{2} with frame bound A=3/2A = 3 / 2.

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Problem 8.2

Show that if {xn}\left\{ x_{n}\right\} is a frame for a Hilbert space HH and {yn}\left\{ y_{n}\right\} is a Bessel sequence in HH, then {xn}∪{yn}\left\{ x_{n}\right\} \cup \left\{ y_{n}\right\} is a frame for HH.

?
Problem 8.3

Show that if {xn}n∈N\left\{ x_{n}\right\}_{n \in \mathbb {N}} is a Riesz basis and J⊆NJ \subseteq \mathbb {N}, then {xn}n∈J\left\{ x_{n}\right\}_{n \in J} is a Riesz basis for span⁡‾{xn}n∈J\overline{\operatorname {span}}\left\{ x_{n}\right\}_{n \in J}. Does the analogous statement hold for frames?

?
Problem 8.4

Let {xn}\left\{ x_{n}\right\} be a sequence in a Hilbert space HH, and let EE be a dense subset of HH. Show that if there exist some A,B>0A, B > 0 such that A∥x∥2≤∑∣⟨x,xn⟩∣2≤B∥x∥2A\left\| x\right\|^{2} \leq \sum \left| \left\langle x, x_{n} \right\rangle \right|^{2} \leq B\left\| x\right\|^{2} for x∈Ex \in E, then {xn}\left\{ x_{n}\right\} is a frame for HH. Thus it suffices to establish the frame condition on some dense (and hopefully "nice") subset of HH (compare hypothesis (b) of Theorem 7.4).

?
Problem 8.5

Let {xn}\left\{ x_{n}\right\} be an AA-tight frame for a Hilbert space HH. Show that the frame operator for {xn}\left\{ x_{n}\right\} (see Definition 7.3) is S=AIS = A I, and use this to show that x=A−1∑⟨x,xn⟩xnx = A^{-1} \sum \left\langle x, x_{n} \right\rangle x_{n} for x∈Hx \in H.

?
Problem 8.6

Let {xn}\left\{ x_{n}\right\} be an AA-tight frame in a Hilbert space HH.

?
(a)

Show that ∥xn∥2≤A\left\| x_{n}\right\|^{2} \leq A for every n∈Nn \in \mathbb {N}.

(b)

Show that if ∥xm∥2<A\left\| x_{m}\right\|^{2} < A for some mm, then {xn}n≠m\left\{ x_{n}\right\}_{n \neq m} is a frame for HH, and the optimal lower frame bound for {xn}n≠m\left\{ x_{n}\right\}_{n \neq m} is A−∥xm∥2A-\left\| x_{m}\right\|^{2} (note that this new frame might not be tight).

(c)

Show that if ∥xm∥2=A\left\| x_{m}\right\|^{2} = A for some mm, then xm⊥xnx_{m} \perp x_{n} for all n≠mn \neq m.

Problem 8.7

Let {xn}\left\{ x_{n}\right\} be an AA-tight frame for a Hilbert space HH. Show directly that the following statements are equivalent.

?
(a)

∥xn∥2=A\left\| x_{n}\right\|^{2} = A for every nn.

(b)

{xn}\left\{ x_{n}\right\} is an orthogonal (but not necessarily orthonormal) sequence with no zero elements.

(c)

{xn}\left\{ x_{n}\right\} is a basis for HH.

(d)

{xn}\left\{ x_{n}\right\} is ω\omega-independent.

(e)

{xn}\left\{ x_{n}\right\} is an exact frame.

Use these equivalences to show that {xn}\left\{ x_{n}\right\} is an exact Parseval frame for HH if and only if it is an orthonormal basis for HH.

Problem 8.8

Let PP be the orthogonal projection of a Hilbert space HH onto a closed subspace MM. Show that if {xn}\left\{ x_{n}\right\} is an orthonormal basis for HH, then {Pxn}\left\{ P x_{n}\right\} is a Parseval frame for MM.

?
Problem 8.9

Given λ∈R\lambda \in \mathbb {R}, let eλ(t)=e2πiλte_{\lambda }(t) = e^{2 \pi i \lambda t} for t∈[0,1)t \in [0,1).

?
(a)

Show that if b>1b > 1, then span⁡‾{ebn}n∈Z\overline{\operatorname {span}}\left\{ e_{b n}\right\}_{n \in \mathbb {Z}} is a proper subspace of L2(T)L^{2}(\mathbf{T}). In particular, find a nonzero function in L2(T)L^{2}(\mathbf{T}) that is orthogonal to ebne_{b n} for every n∈Zn \in \mathbb {Z}.

(b)

Show that if 0<b<10 < b < 1 then {ebn}n∈Z\left\{ e_{b n}\right\}_{n \in \mathbb {Z}} is a tight frame for L2(T)L^{2}(\mathbf{T}), and find the frame bound. Show directly that this sequence is not orthogonal, and demonstrate that it is not a basis by finding two distinct ways to write the constant function as 1=∑cnebn1 = \sum c_{n} e_{b n}, where the series converges in L2L^{2}-norm. Show that {ebn}n≠0\left\{ e_{b n}\right\}_{n \neq 0} is a frame for L2(T)L^{2}(\mathbf{T}), and therefore {ebn}n∈Z\left\{ e_{b n}\right\}_{n \in \mathbb {Z}} is inexact.

Problem 8.10
?
(a)

Prove the following perturbation result for frames. Suppose that {xn}\left\{ x_{n}\right\} is a frame for a Hilbert space HH with frame bounds A,BA, B, and {yn}⊆H\left\{ y_{n}\right\} \subseteq H is such that {fn−gn}\left\{ f_{n}-g_{n}\right\} is a Bessel sequence with Bessel bound KK. Show that {gn}\left\{ g_{n}\right\} is a frame if K<AK < A.

(b)

Show that if {hn}\left\{ h_{n}\right\} is a sequence in HH that satisfies K=∑∥hn∥2<∞K = \sum \left\| h_{n}\right\|^{2} < \infty, then {hn}\left\{ h_{n}\right\} is a Bessel sequence with Bessel bound KK.

(c)

Exercise 8.9 showed that {e2πibnt}n∈Z\left\{ e^{2 \pi i b n t}\right\}_{n \in \mathbb {Z}} is a frame for L2(T)L^{2}(\mathbf{T}) when 0<b≤10 < b \leq 1. Combine this with parts (a) and (b) of this exercise to formulate and prove a theorem establishing a sufficient condition on numbers λn∈R\lambda_{n} \in \mathbb {R} so that {e2πiλnt}n∈Z\left\{ e^{2 \pi i \lambda_{n} t}\right\}_{n \in \mathbb {Z}} is a frame for L2(T)L^{2}(\mathbf{T}). Do you think your result is optimal?

Problem 8.11

Let v1,…,vnv_{1}, \ldots , v_{n} be vectors in Fd\mathbf{F}^{d}. Give direct proofs of the following statements.

?
(a)

{v1,…,vn}\left\{ v_{1}, \ldots , v_{n}\right\} is a Bessel sequence in Fd\mathbf{F}^{d}, its synthesis operator corresponds to multiplication by the matrix RR that has v1,…,vnv_{1}, \ldots , v_{n} as columns, and its analysis operator is C=R∗C = R^{*}, the Hermitian of the matrix RR.

(b)

The following are equivalent: (i) {v1,…,vn}\left\{ v_{1}, \ldots , v_{n}\right\} spans Fd\mathbf{F}^{d}, (ii) S=C∗CS = C^{*} C is positive definite, (iii) {v1,…,vn}\left\{ v_{1}, \ldots , v_{n}\right\} is a frame for Fd\mathbf{F}^{d}. Further, in case these hold, the optimal frame bounds for {v1,…,vn}\left\{ v_{1}, \ldots , v_{n}\right\} are λ1,λd\lambda_{1}, \lambda_{d}, where λ1\lambda_{1} is the smallest eigenvalue of SS and λd\lambda_{d} is the largest eigenvalue.

(c)

The following are equivalent: (i) {v1,…,vn}\left\{ v_{1}, \ldots , v_{n}\right\} is linearly independent, (ii) G=CC∗G = C C^{*} is positive definite, (iii) {v1,…,vn}\left\{ v_{1}, \ldots , v_{n}\right\} is a Riesz sequence in Fd\mathbf{F}^{d}.

Problem 8.12

Let {v1,…,vN}\left\{ v_{1}, \ldots , v_{N}\right\} be a finitely many vectors in a Hilbert space HH.

?
(a)

Show that {v1,…,vN}\left\{ v_{1}, \ldots , v_{N}\right\} is a frame sequence, i.e., it is a frame for span⁡{v1,…,vN}\operatorname {span}\left\{ v_{1}, \ldots , v_{N}\right\}.

(b)

Assume {v1,…,vN}\left\{ v_{1}, \ldots , v_{N}\right\} is linearly independent, and let A,BA, B be frame bounds for {v1,…,vN}\left\{ v_{1}, \ldots , v_{N}\right\} as a frame for its span. Show that

∀c1,…,cN,A∑k=1N∣ck∣2≤∥∑k=1Nckvk∥2≤B∑k=1N∣ck∣2 \forall c_{1}, \ldots , c_{N}, \quad A \sum _{k = 1}^{N}\left|c_{k}\right|^{2} \leq \left\| \sum _{k = 1}^{N} c_{k} v_{k}\right\| ^{2} \leq B \sum _{k = 1}^{N}\left|c_{k}\right|^{2}
Problem 8.13

A sequence {xn}\left\{ x_{n}\right\} in a Banach space XX is called a quasibasis if there exist an∈X∗a_{n} \in X^{*} such that x=∑⟨x,an⟩xnx = \sum \left\langle x, a_{n} \right\rangle x_{n} for all x∈Xx \in X. All Schauder bases and frames are therefore quasibases. Exhibit a quasibasis for a Hilbert space that is neither a basis nor a frame.

?
Problem 8.14

Given a sequence {xn}\left\{ x_{n}\right\} in a Hilbert space HH, show that {xn}\left\{ x_{n}\right\} is a frame with frame bounds A,BA, B if and only if Sx=∑⟨x,xn⟩xnS x = \sum \left\langle x, x_{n} \right\rangle x_{n} is a well-defined positive linear mapping of HH into HH that satisfies AI≤S≤BIA I \leq S \leq B I.

?
Problem 8.15

This exercise will give an alternative proof that the frame operator SS is a topological isomorphism.

?
(a)

Show that if U,V∈B(H)U, V \in \mathcal{B}(H) are positive and U≤VU \leq V then ∥U∥≤∥V∥\left\| U\right\| \leq \left\| V\right\|.

(b)

Let SS be the frame operator for a frame {xn}\left\{ x_{n}\right\} that has frame bounds A,BA, B. Prove the operator inequalities

0≤I−1BS≤B−ABI 0 \leq I-\frac{1}{B} S \leq \frac{B-A}{B} I

and

0≤I−2A+BS≤B−AB+AI 0 \leq I-\frac{2}{A+B} S \leq \frac{B-A}{B+A} I

and use either one of these to show that SS is a topological isomorphism.

Problem 8.16

Prove Theorem 8.18: let {xn}\left\{ x_{n}\right\} be a frame for HH with analysis operator CC.

?
(a)

If {yn}\left\{ y_{n}\right\} is a Bessel sequence in HH that is an alternative dual for {xn}\left\{ x_{n}\right\}, then {yn}\left\{ y_{n}\right\} is a frame and {xn}\left\{ x_{n}\right\} is an alternative dual frame for {yn}\left\{ y_{n}\right\}.

(b)

A Bessel sequence {yn}\left\{ y_{n}\right\} is the canonical dual frame for {xn}\left\{ x_{n}\right\} if and only if it is an alternative dual and range⁡(C)=range⁡(C~)\operatorname {range}(C) = \operatorname {range}(\widetilde{C}), where C~\widetilde{C} is the analysis operator of {yn}\left\{ y_{n}\right\}.

Problem 8.17

Complete the proof of Theorem 8.19: let MM be a closed subspace of HH, let {xn}\left\{ x_{n}\right\} be a frame for HH, and let PP be the orthogonal projection of HH onto MM.

?
(a)

{Pxn}\left\{ P x_{n}\right\} is a frame for MM with the same frame bounds as {xn}\left\{ x_{n}\right\}, and {Px~n}\left\{ P \widetilde{x}_{n}\right\} is an alternative dual frame for {Pxn}\left\{ P x_{n}\right\}.

(b)

{Px~n}\left\{ P \widetilde{x}_{n}\right\} is the canonical dual frame for {Pxn}\left\{ P x_{n}\right\} if and only if PS=SPPS = SP, where SS is the frame operator for {xn}\left\{ x_{n}\right\}.

Problem 8.18

Let {xn}\left\{ x_{n}\right\} be a frame for a Hilbert space HH. Let C,R,SC, R, S be the analysis, synthesis, and frame operators for {xn}\left\{ x_{n}\right\}, and let C~,R~\widetilde{C}, \widetilde{R} be the analysis and synthesis operators for the canonical dual frame {x~n}\left\{ \widetilde{x}_{n}\right\}.

?
(a)

Show that C~=CS−1\widetilde{C} = C S^{-1} and R~=S−1R\widetilde{R} = S^{-1} R.

(b)

Show that the orthogonal projection PP of ℓ2\ell^{2} onto range (C)(C) is Pc=CR~c=CS−1Rc={⟨∑ncnx~n,xk⟩}k∈N,c=(cn)∈ℓ2P c = C \widetilde{R} c = C S^{-1} R c = \left\{ \left\langle \sum_{n} c_{n} \widetilde{x}_{n}, x_{k} \right\rangle \right\}_{k \in \mathbb {N}}, \quad c = \left(c_{n}\right) \in \ell^{2}.

Problem 8.19

Let {xn}\left\{ x_{n}\right\} be a frame for a Hilbert space HH and let C:H→ℓ2C: H \rightarrow \ell^{2} be its analysis operator. Let {δn}\left\{ \delta_{n}\right\} be the standard basis for ℓ2\ell^{2}, and prove that the following two statements are equivalent.

?
(a)

{yn}\left\{ y_{n}\right\} is an alternative dual frame for {xn}\left\{ x_{n}\right\}.

(b)

There is a bounded left-inverse VV of CC such that yn=Vδny_{n} = V \delta_{n} for all n∈Nn \in \mathbb {N} (that is, V:ℓ2→HV: \ell^{2} \rightarrow H is bounded, VC=IV C = I, and yn=Vδny_{n} = V \delta_{n}).

Problem 8.20

Let T∈B(H)T \in \mathcal{B}(H) be a Hilbert-Schmidt operator on a Hilbert space HH (see Definition B.7). Show that if {xn}\left\{ x_{n}\right\} is a frame for HH with canonical dual frame {x~n}\left\{ \widetilde{x}_{n}\right\}, then the Hilbert-Schmidt norm of TT is given by ∥T∥HS2=∑⟨Txn,Tx~n⟩\left\| T\right\|_{\mathrm{HS}}^{2} = \sum \left\langle T x_{n}, T \widetilde{x}_{n} \right\rangle.

?
Problem 8.21

Prove Corollary 8.23.

?
Remark.
?

Corollary 8.23: If {xn}\left\{ x_{n}\right\} is a frame for a Hilbert space HH, then the following statements are equivalent. (a) {xn}\left\{ x_{n}\right\} is an exact frame. (b) {xn}\left\{ x_{n}\right\} and {x~n}\left\{ \widetilde{x}_{n}\right\} are biorthogonal. (c) ⟨xn,x~n⟩=1\left\langle x_{n}, \widetilde{x}_{n} \right\rangle = 1 for all nn.

Problem 8.22

Prove Corollary 8.24: let {xn}\left\{ x_{n}\right\} be a frame for HH with frame bounds A,BA, B.

?
(a)

{xn}\left\{ x_{n}\right\} is norm-bounded above, with ∥xn∥2≤B\left\| x_{n}\right\|^{2} \leq B for every nn.

(b)

If {xn}\left\{ x_{n}\right\} is exact then it is norm-bounded below, with A≤∥xn∥2A \leq \left\| x_{n}\right\|^{2} for every nn.

Problem 8.23

Show that if {xn}\left\{ x_{n}\right\} is an exact frame for a Hilbert space HH then it has a unique dual frame (the canonical dual).

?
Problem 8.24

Prove Lemma 8.26.

?
Remark.
?

Lemma 8.26: Frames are preserved by topological isomorphisms. Specifically, if {xn}\left\{ x_{n}\right\} is a frame for a Hilbert space HH and T:H→KT: H \rightarrow K is a topological isomorphism, then {Txn}\left\{ T x_{n}\right\} is a frame for KK and the following additional statements hold as well. (a) If A,BA, B are frame bounds for {xn}\left\{ x_{n}\right\}, then {Txn}\left\{ T x_{n}\right\} has frame bounds A∥T−1∥−2,B∥T∥2A\left\| T^{-1}\right\|^{-2}, B\left\| T\right\|^{2}. (b) If SS is the frame operator for {xn}\left\{ x_{n}\right\}, then {Txn}\left\{ T x_{n}\right\} has frame operator TST∗T S T^{*}. (c) {xn}\left\{ x_{n}\right\} is exact if and only if {Txn}\left\{ T x_{n}\right\} is exact.

Problem 8.25

Let {xn}\left\{ x_{n}\right\} be a frame for a Hilbert space HH. Show that if {xn}\left\{ x_{n}\right\} is a Riesz basis for HH and MM is a proper closed subspace of HH, then {Pxn}\left\{ P x_{n}\right\} is a frame but is not a Riesz basis for MM, where PP denotes the orthogonal projection of HH onto MM.

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Problem 8.26

Let {xn}\left\{ x_{n}\right\} be a frame for a Hilbert space HH. Let C,RC, R be the analysis and synthesis operators for {xn}\left\{ x_{n}\right\}, and let C~,R~\widetilde{C}, \widetilde{R} be the analysis and synthesis operators for the canonical dual frame {x~n}\left\{ \widetilde{x}_{n}\right\}.

?
(a)

Show that range⁡(C)=range⁡(C~)\operatorname {range}(C) = \operatorname {range}(\widetilde{C}).

(b)

Show that the pseudoinverse R†R^{\dagger } of the synthesis operator RR is R†=C~R^{\dagger } = \widetilde{C}, so

R†x=C~x={⟨x,x~n⟩}={⟨x,S−1xn⟩},x∈H R^{\dagger } x = \widetilde{C} x = \left\{ \left\langle x, \widetilde{x}_{n} \right\rangle \right\} = \left\{ \left\langle x, S^{-1} x_{n} \right\rangle \right\} , \quad x \in H
(c)

Show that the optimal frame bounds for {xn}\left\{ x_{n}\right\} are

A=∥C~∥−2=∥S−1∥−1 and B=∥C∥2=∥S∥ A = \left\| \widetilde{C}\right\| ^{-2} = \left\| S^{-1}\right\| ^{-1} \quad \text{ and } \quad B = \left\| C\right\| ^{2} = \left\| S\right\|
Problem 8.27

Prove Theorem 8.32.

?
Remark.
?

Theorem 8.32: Given a sequence {xn}\left\{ x_{n}\right\} in a Hilbert space HH, the following statements are equivalent. (a) {xn}\left\{ x_{n}\right\} is a Riesz basis for HH. (b) The analysis operator Cx={⟨x,xn⟩}C x = \left\{ \left\langle x, x_{n} \right\rangle \right\} maps HH bijectively onto ℓ2\ell^{2}. (c) The synthesis operator Rc=∑cnxnR c = \sum c_{n} x_{n} is well defined for each c=(cn)∈ℓ2c = (c_{n}) \in \ell^{2}, and maps ℓ2\ell^{2} bijectively onto HH. (d) There exists an orthonormal basis {en}\left\{ e_{n}\right\} for HH and a bounded bijection T∈B(H)T \in \mathcal{B}(H) such that Ten=xnT e_{n} = x_{n} for every nn. (e) {xn}\left\{ x_{n}\right\} is an ω\omega-independent frame. (f) {xn}\left\{ x_{n}\right\} is an ℓ2\ell^{2}-independent frame, i.e., it is a frame and if ∑cnxn=0\sum c_{n} x_{n} = 0 for some sequence (cn)∈ℓ2(c_{n}) \in \ell^{2} then cn=0c_{n} = 0 for every nn.

Problem 8.28

Let {xn}\left\{ x_{n}\right\} be a Bessel sequence in a Hilbert space HH, and let R:ℓ2→HR: \ell^{2} \rightarrow H be its synthesis operator.

?
(a)

Show that {xn}\left\{ x_{n}\right\} is complete if and only if RR has dense range.

(b)

Show that {xn}\left\{ x_{n}\right\} is ℓ2\ell^{2}-independent if and only if RR is injective.

(c)

Now assume that {xn}\left\{ x_{n}\right\} is a Bessel sequence that is a conditional Schauder basis for HH (see Exercise 7.1 for an example). Show that there exists some x∈Hx \in H such that (⟨x,x~n⟩)∉ℓ2\left( \left\langle x, \widetilde{x}_{n} \right\rangle \right) \notin \ell^{2}, where (x~n)\left(\widetilde{x}_{n}\right) is the dual basis to {xn}\left\{ x_{n}\right\}. Show that the sequence {xn}∪{x}\left\{ x_{n}\right\} \cup \left\{ x\right\} is ℓ2\ell^{2}-independent but not ω\omega-independent.

Problem 8.29

Prove the remaining statements in Corollary 8.33.

?
(a)

{xn}\left\{ x_{n}\right\} is a frame for HH if and only if there exists a closed subspace M⊆ℓ2M \subseteq \ell^{2} and a topological isomorphism T:M→HT: M \rightarrow H such that xn=TPMδnx_{n} = T P_{M} \delta_{n} for every nn.

(b)

{xn}\left\{ x_{n}\right\} is a Parseval frame for HH if and only if the operator TT in part (a) can be taken to be an isometric isomorphism.

Problem 8.30

Prove Corollary 8.34 (Naimark Duality).

?
(a)

{xn}\left\{ x_{n}\right\} is a frame for HH if and only if there exists a Hilbert space K⊇HK \supseteq H and a Riesz basis {en}\left\{ e_{n}\right\} for KK such that PHen=xnP_{H} e_{n} = x_{n} for each nn.

(b)

{xn}\left\{ x_{n}\right\} is a Parseval frame for HH if and only if there exists a Hilbert space K⊇HK \supseteq H and an orthonormal basis {en}\left\{ e_{n}\right\} for KK such that PHen=xnP_{H} e_{n} = x_{n} for each nn.

Problem 8.31
?
(a)

Exhibit a frame for Fd\mathbf{F}^{d} that contains infinitely many nonzero vectors.

(b)

Show that if {xn}n∈I\left\{ x_{n}\right\}_{n \in I} is a frame for Fd\mathbf{F}^{d} and inf⁡n∈I∥xn∥>0\inf_{n \in I}\left\| x_{n}\right\| > 0, then II is finite.

Problem 8.32

Let {xn}\left\{ x_{n}\right\} be a frame for a Hilbert space HH and let {yn}\left\{ y_{n}\right\} be a frame for a Hilbert space KK. Inspired by Definition 4.19, declare {xn}\left\{ x_{n}\right\} and {yn}\left\{ y_{n}\right\} to be equivalent if there exists a topological isomorphism T:H→KT: H \rightarrow K such that Txn=ynT x_{n} = y_{n} for every nn.

?
(a)

Show that {xn}\left\{ x_{n}\right\} and {yn}\left\{ y_{n}\right\} are equivalent if and only if range (CX)=\left(C_{X}\right) = range (CY)\left(C_{Y}\right), where CX,CYC_{X}, C_{Y} are the analysis operators for {xn}\left\{ x_{n}\right\} and {yn}\left\{ y_{n}\right\}.

(b)

Show that part (a) is equivalent to:

∀(cn)∈ℓ2,∑cnxn=0⟺∑cnyn=0. \forall \left(c_{n}\right) \in \ell ^{2}, \quad \sum c_{n} x_{n} = 0 \Longleftrightarrow \sum c_{n} y_{n} = 0.
(c)

Exhibit a frame {xn}\left\{ x_{n}\right\} and a permutation σ\sigma of N\mathbb {N} that fixes all but finitely many elements of N\mathbb {N} such that {xn}\left\{ x_{n}\right\} is not equivalent to {xσ(n)}\left\{ x_{\sigma (n)}\right\}. Show that this cannot happen if {xn}\left\{ x_{n}\right\} is an exact frame.

(d)

Exhibit a frame {xn}\left\{ x_{n}\right\} and a sequence of signs εn=±1\varepsilon_{n} = \pm 1 such that {xn}\left\{ x_{n}\right\} is not equivalent to {εnxn}\left\{ \varepsilon_{n} x_{n}\right\}. Show that this cannot happen if {xn}\left\{ x_{n}\right\} is an exact frame.

Remark Parts (c) and (d) illustrate some of the weaknesses of this notion of equivalence for frames. A more robust notion of frame equivalence was introduced by Balan and Landau in [BL07].

Problem 8.33

Show that {xn}\left\{ x_{n}\right\} is a Parseval frame for a Hilbert space HH if and only if there exists a Hilbert space KK and a Parseval frame {yn}\left\{ y_{n}\right\} for KK such that {(xn,yn)}\left\{ \left(x_{n}, y_{n}\right)\right\} is an orthonormal basis for H×KH \times K.

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Problem 8.34

Let HH be an infinite-dimensional separable Hilbert space. Show that if MM is an infinite-dimensional closed subspace of ℓ2\ell^{2}, then there exists a frame {xn}\left\{ x_{n}\right\} for HH such that range⁡(C)=M\operatorname {range}(C) = M, where CC is the analysis operator.

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Problem 8.35

This exercise is about superframes as introduced by Balan [Bal98] and Han and Larson [HL00]. Let {xn}\left\{ x_{n}\right\} be a frame for a Hilbert space HH and let {yn}\left\{ y_{n}\right\} be a frame for a Hilbert space KK. Let CXC_{X} be the analysis operator for {xn}\left\{ x_{n}\right\} and CYC_{Y} the analysis operator for {yn}\left\{ y_{n}\right\}.

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(a)

Show that if range⁡(CX)⊥range⁡(CY)\operatorname {range}\left(C_{X}\right) \perp \operatorname {range}\left(C_{Y}\right) then {(xn,yn)}\left\{ \left(x_{n}, y_{n}\right)\right\} is a frame for H×KH \times K, and it is a Parseval frame if and only if both {xn}\left\{ x_{n}\right\} and {yn}\left\{ y_{n}\right\} are Parseval. (When range⁡(CX)⊥range⁡(CY)\operatorname {range}\left(C_{X}\right) \perp \operatorname {range}\left(C_{Y}\right), we say that {xn}\left\{ x_{n}\right\} and {yn}\left\{ y_{n}\right\} are orthogonal frames.)

(b)

Suppose that, in addition, we have range⁡(CX)⊕range⁡(CY)=ℓ2\operatorname {range}\left(C_{X}\right) \oplus \operatorname {range}\left(C_{Y}\right) = \ell^{2}. Show that {(xn,yn)}\left\{ \left(x_{n}, y_{n}\right)\right\} is a Riesz basis for H×KH \times K, and it is an orthonormal basis if and only if {xn}\left\{ x_{n}\right\} and {yn}\left\{ y_{n}\right\} are each Parseval frames (in this case, we call {(xn,yn)}\left\{ \left(x_{n}, y_{n}\right)\right\} a superframe).

(c)

Give examples of frames {xn}\left\{ x_{n}\right\} and {yn}\left\{ y_{n}\right\} that are not Riesz bases but are such that {(xn,yn)}\left\{ \left(x_{n}, y_{n}\right)\right\} is a Riesz basis for H×HH \times H.

(d)

Given a frame {xn}\left\{ x_{n}\right\} for HH, show there exists a frame {yn}\left\{ y_{n}\right\} for HH such that {(xn,yn)}\left\{ \left(x_{n}, y_{n}\right)\right\} is a Riesz basis for H×HH \times H.

Problem 8.36

This exercise gives another way to construct frames for "larger" spaces from "smaller" ones.

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(a)

Let {xn},{yn}\left\{ x_{n}\right\} ,\left\{ y_{n}\right\} be frames for Hilbert spaces H,KH, K, respectively. Show that the tensor product sequence {xm⊗yn}m,n∈N\left\{ x_{m} \otimes y_{n}\right\}_{m, n \in \mathbb {N}} is a frame for H⊗K=B2(H,K)H \otimes K = \mathcal{B}_{2}(H, K) (see Appendix B for definitions).

(b)

Let E,FE, F be measurable subsets of R\mathbb {R}. Show that if {fn}\left\{ f_{n}\right\} is a frame for L2(E)L^{2}(E) and {gn}\left\{ g_{n}\right\} is a frame for L2(F)L^{2}(F), then {fm(x)‾gn(y)}m,n∈N\left\{ \overline{f_{m}(x)} g_{n}(y)\right\}_{m, n \in \mathbb {N}} is a frame for L2(E×F)L^{2}(E \times F).

Problem 8.37

Complete the proof of Theorem 8.39.

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Remark.
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Theorem 8.39: Given a frame {xn}\left\{ x_{n}\right\} for a Hilbert space HH, the following statements are equivalent. (a) {xn}\left\{ x_{n}\right\} is a near-Riesz basis. (b) {xn}\left\{ x_{n}\right\} is Besselian. (c) range⁡(C)⊥=ker⁡(R)\operatorname {range}(C)^{\perp } = \ker (R) is finite dimensional. Moreover, if {xn}\left\{ x_{n}\right\} is norm-bounded below, then statements (a)--(c) are also equivalent to the following statement. (d) {xn}\left\{ x_{n}\right\} is an unconditional frame.

Problem 8.38

Prove that the matrix appearing on the left-hand side of equation (8.9) has column rank NN, i.e., its columns span an NN-dimensional subspace of FN\mathbf{F}^{N}.

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Problem 8.39

Verify the claim in the proof of Theorem 8.44 that there exists a subsequence such that equation (8.10) is satisfied.

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Problem 8.40

Let F={xn}\mathcal{F} = \left\{ x_{n}\right\} be a frame for a Hilbert space HH with frame bounds A,BA, B and canonical dual frame {x~n}\left\{ \widetilde{x}_{n}\right\}. Let G={xnk}\mathcal{G} = \left\{ x_{n_{k}}\right\} be a subsequence of {xn}\left\{ x_{n}\right\}, and show that the following statements are equivalent.

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(a)

{xn}n≠nk\left\{ x_{n}\right\}_{n \neq n_{k}} is complete (and hence a frame) for each k∈Nk \in \mathbb {N}, and there exists a single constant L>0L > 0 that is a lower frame bound for each frame {xn}n≠nk\left\{ x_{n}\right\}_{n \neq n_{k}}.

(b)

sup⁡k⟨xnk,x~nk⟩<1\sup_{k} \left\langle x_{n_{k}}, \widetilde{x}_{n_{k}} \right\rangle < 1.

Show further that if these statements hold, then there exists an infinite subsequence Gε\mathcal{G}_{\varepsilon } of G\mathcal{G} such that F\Gε\mathcal{F} \backslash \mathcal{G}_{\varepsilon } is a frame for HH with frame bounds L(A/B)−ε,1L(A / B)-\varepsilon , 1.

Remark It is shown in [BCHL03] that Gε\mathcal{G}_{\varepsilon } can be chosen so that F\Gε\mathcal{F} \backslash \mathcal{G}_{\varepsilon } has frame bounds L−ε,1L-\varepsilon , 1.

Problem 8.41

Prove Lemma 8.47: let {xn}\left\{ x_{n}\right\} be a Bessel sequence with analysis operator CC and synthesis operator R=C∗R = C^{*}, and suppose U,VU, V are operators such that VC=CUV C = C U.

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(a)

U(ker⁡(C))⊆ker⁡(C)U\left(\operatorname {ker}(C)\right) \subseteq \operatorname {ker}(C).

(b)

V∗(ker⁡(R))⊆ker⁡(R)V^{*}\left(\operatorname {ker}(R)\right) \subseteq \operatorname {ker}(R).

(c)

V(range⁡(C))⊆range⁡(C)V\left(\operatorname {range}(C)\right) \subseteq \operatorname {range}(C).

(d)

U∗(range⁡(R))⊆range⁡(R)U^{*}\left(\operatorname {range}(R)\right) \subseteq \operatorname {range}(R).

Problem 8.42
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(a)

Prove that the operators U,V,U∗U, V, U^{*}, and V∗V^{*} appearing in Example 8.49 have no eigenvalues.

(b)

Show that if 0<b<10 < b < 1, then there is an infinite set J⊆ZJ \subseteq \mathbb {Z} such that {e2πibnt}n∉J\left\{ e^{2 \pi i b n t}\right\}_{n \notin J} is a frame for L2(T)L^{2}(\mathbf{T}).