Math Problems Vault

Problem 2.1

Let $M$ be a subspace of a Hilbert space $H$ and fix $\lambda \in M$ . Show directly that there exists some $\Lambda \in H$ such that $\left\langle x, \Lambda \right\rangle = \left\langle x, \lambda \right\rangle$ for all $x \in M$ and $\left\| \Lambda \right\| = \left\| \lambda \right\|$ .

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Problem 2.2

Suppose that $M$ is a closed subspace of a Hilbert space $H, x_{0} \in H \backslash M$ , and $d = \operatorname {dist}\left(x_{0}, M\right)$ . Show directly that there exists a $\mu \in H$ such that $\left\langle x_{0}, \mu \right\rangle = 1, \left\langle x, \mu \right\rangle = 0$ for all $x \in M$ , and $\left\| \mu \right\| = 1 / d$ .

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Problem 2.3

(a)(b)

Let $X$ be a normed space.

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(a)

Show that if $X^{*}$ is separable then $X$ is separable.

(b)

Show that the converse can fail.

Problem 2.4

The Weierstrass Approximation Theorem implies that $\left\{ x^{k}\right\}_{k \geq 0}$ is complete in $C[0,1]$ . Show that $\left\{ x^{2 k}\right\}_{k \geq 0}$ is also complete in $C[0,1]$ .

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Problem 2.5

Given a subset $A$ of a normed space $X$ , define its orthogonal complement $A^{\perp } \subseteq X^{*}$ by

A^{\perp } = \left\{ \mu \in X^{*}: \left\langle x, \mu \right\rangle = 0 \text{ for all } x \in A\right\}

Prove that $A^{\perp }$ is a closed subspace of $X^{*}$ , and explain how this relates to Corollary 2.5.

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Problem 2.6

Let $S$ be a subspace of a normed space $X$ , and show that its closure $\bar{S}$ is given by

\bar{S} = \bigcap \left\{ \operatorname {ker}(\mu ): \mu \in X^{*} \text{ and } S \subseteq \operatorname {ker}(\mu )\right\} .

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Problem 2.7

Show that $\ell^{p}$ is reflexive for each $1 < p < \infty$ .

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Problem 2.8

Let $X$ be a Banach space. Show that if $X$ is separable but $X^{*}$ is not, then $X$ is not reflexive. Use this to show that $\ell^{1}$ is a proper subspace of $\left(\ell^{\infty }\right)^{*}$ .

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Problem 2.9

Let $X, Y$ be Banach spaces. Given $T \in \mathcal{B}(X, Y)$ , show that there is a unique operator $T^{*} \in \mathcal{B}\left(Y^{*}, X^{*}\right)$ that satisfies equation (2.4).

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Problem 2.10

Let $X$ be a Banach space. Given $\mu \in X^{*} = \mathcal{B}(X, \mathbf{F})$ , explicitly describe its adjoint $\mu^{*}$ .

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Problem 2.11

Let $M$ be a closed subspace $M$ of a normed space $X$ , and fix $L \in \mathcal{B}(X)$ . We say that $M$ is invariant under $L$ if $L(M) \subseteq M$ . Show that if $M \subseteq X$ is invariant under $L$ , then $M^{\perp }$ is invariant under $L^{*}$ , where $M^{\perp }$ is the orthogonal complement defined in Exercise 2.5.

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Problem 2.12

Suppose that $M$ is a closed subspace of a Banach space $X$ . Let $\epsilon : M \rightarrow X$ be the embedding map, i.e., $\epsilon (x) = x$ for $x \in M$ . Show that $\epsilon^{*}: X^{*} \rightarrow M^{*}$ is the restriction map, i.e., if $\mu \in X^{*}$ , then $\epsilon^{*} \mu = \left.\mu \right|_{M}$ .

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Problem 2.13

Let $A: \mathbb {C}^{n} \rightarrow \mathbb {C}^{m}$ be a linear operator, which we identify with its $m \times n$ matrix representation. Show that the adjoint of $A$ in the Hilbert space sense (Definition 2.10) is the conjugate transpose matrix $A^{*} = \overline{A^{\mathrm{T}}}$ , while the adjoint of $A$ in the Banach space sense (Definition 2.8) is the transpose matrix $A^{\mathrm{T}}$ .

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Problem 2.14

Let $L, R$ be the left- and right-shift operators on $\ell^{2}$ defined in Exercise 1.64. Show that $R = L^{*}$ .

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Problem 2.15

Fix $\lambda \in \ell^{\infty }$ , and let $M_{\lambda }$ be the multiplication operator defined in Exercise 1.66. Find $M_{\lambda }^{*}$ , and determine when $M_{\lambda }$ is self-adjoint, positive, or positive definite.

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Problem 2.16

Prove Theorem 2.13.

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Remark.

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Theorem 2.13: Let $H, K, L$ be Hilbert spaces, and fix $A \in \mathcal{B}(H, K)$ and $B \in \mathcal{B}(K, L)$ . (a) $(A^{*})^{*} = A$ . (b) $(B A)^{*} = A^{*} B^{*}$ . (c) $\ker (A) = \operatorname {range}(A^{*})^{\perp }$ . (d) $\ker (A)^{\perp } = \overline{\operatorname {range}(A^{*})}$ . (e) $A$ is injective if and only if $\operatorname {range}(A^{*})$ is dense in $H$ . (f) $\left\| A\right\| = \left\| A^{*}\right\| = \left\| A^{*} A\right\|^{1 / 2} = \left\| A A^{*}\right\|^{1 / 2}$ .

Problem 2.17

Let $M$ be a closed subspace of a Hilbert space $H$ , and let $P \in \mathcal{B}(H)$ be given. Show that $P$ is the orthogonal projection of $H$ onto $M$ if and only if $P^{2} = P, P^{*} = P$ , and range $(P) = M$ .

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Problem 2.18

(a)(b)(c)

Let $H$ be a Hilbert space and suppose that $A, B \in \mathcal{B}(H)$ are self-adjoint.

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(a)

Show that $A B A$ and $B A B$ are self-adjoint.

(b)

Show that $A B$ is self-adjoint if and only if $A B = B A$ .

(c)

Exhibit self-adjoint operators $A, B$ that do not commute.

Problem 2.19

(a)(b)(c)

Let $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be fixed.

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(a)

Show that if $A$ is self-adjoint then all eigenvalues of $A$ are real, and eigenvectors of $A$ corresponding to distinct eigenvalues are orthogonal.

(b)

Show that if $A$ is a positive operator then all eigenvalues of $A$ are real and nonnegative.

(c)

Show that if $A$ is a positive definite operator then all eigenvalues of $A$ are real and strictly positive.

Problem 2.20

Let $H, K$ be Hilbert spaces. Show that if $A \in \mathcal{B}(H, K)$ , then $A^{*} A \in \mathcal{B}(H)$ and $A A^{*} \in \mathcal{B}(K)$ are positive operators.

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Problem 2.21

(a)(b)

Let $H$ be a Hilbert space. Given $A \in \mathcal{B}(H)$ ,

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(a)

show that $\operatorname {ker}(A) = \operatorname {ker}\left(A^{*} A\right)$ .

(b)

show that $\overline{\operatorname {range}\left(A^{*} A\right)} = \overline{\operatorname {range}\left(A^{*}\right)}$ .

Problem 2.22

Let $H, K$ be Hilbert spaces, and fix $U \in \mathcal{B}(H, K)$ . Show that $U$ is unitary if and only if $U$ is a bijection and $U^{-1} = U^{*}$ .

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Problem 2.23

Fill in the details in the proof of Theorem 2.18.

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(a)

Show that $\left\| A\right\| \leq 1$ when $A$ is self-adjoint with $0 \leq A \leq I$ .

(b)

Show that for $0 < t < \left\| A\right\|^{-1}$ , $t A \leq I$ ; hence $A$ has a square root if and only if $t A$ does.

(c)

Show by induction that $T_{n+1}-T_{n}$ is a polynomial in $B = I-A$ with all nonnegative coefficients.

(d)

Show that $\left\| T_{n}\right\| \leq 1$ for every $n$ , so that $\left( \left\langle T_{n} x, x \right\rangle \right)$ is a bounded, increasing sequence.

(e)

Show that the limit operator $T$ (where $T x = \lim_{n \rightarrow \infty } T_{n} x$ ) is linear and bounded (hence positive).

Problem 2.24

(a)(b)

Let $A$ be a positive operator on a Hilbert space $H$ .

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(a)

Show that $\left\langle A x, x \right\rangle = 0$ if and only if $A x = 0$ .

(b)

Show that the operator $A^{1 / 2}$ constructed in Theorem 2.18 is unique, i.e., there is only one positive operator $S$ satisfying $S^{2} = A$ .

Problem 2.25

Prove Lemma 2.20.

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Remark.

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Lemma 2.20: Let $E$ be a nonempty subset of a Banach space $X$ . Then $E$ is nowhere dense if and only if $E$ contains no nonempty open subsets.

Problem 2.26

Show that $C_{c}(\mathbb {R})$ is a meager subset of $C_{0}(\mathbb {R})$ .

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Problem 2.27

Suppose that $f$ is an infinitely differentiable function on $\mathbb {R}$ such that for each $t \in \mathbb {R}$ there exists some integer $n_{t} \geq 0$ so that $f^{\left(n_{t}\right)}(t) = 0$ . Prove that there exists some open interval $(a, b)$ and some polynomial $p$ such that $f(t) = p(t)$ for all $t \in (a, b)$ .

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Problem 2.28

Let $D$ be the subset of $C[0,1]$ consisting of all functions $f \in C[0,1]$ that have a right-hand derivative at at least one point in $[0,1]$ . Show that $D$ is meager in $C[0,1]$ , and conclude that there are functions in $C[0,1]$ that are not differentiable at any point.

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Problem 2.29

Prove Theorem 2.23.

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Remark.

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Theorem 2.23 (Banach--Steinhaus Theorem): Let $X$ and $Y$ be Banach spaces. If $A_{n} \in \mathcal{B}(X, Y)$ for $n \in \mathbb {N}$ and $A x = \lim_{n \rightarrow \infty } A_{n} x$ exists for each $x \in X$ , then $A \in \mathcal{B}(X, Y)$ and $\left\| A\right\| \leq \sup_{n}\left\| A_{n}\right\| < \infty$ .

Problem 2.30

Let $\left\{ e_{n}\right\}$ be an orthonormal basis for a Hilbert space $H$ , and let $P_{N}$ be the orthogonal projection of $H$ onto $\operatorname {span}\left\{ e_{1}, \ldots , e_{N}\right\}$ . Show that $P_{N} x \rightarrow x$ for every $x \in H$ , but $\left\| I-P_{N}\right\| \nrightarrow 0$ as $N \rightarrow \infty$ .

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Problem 2.31

(a)(b)

Let $X, Y$ be Banach spaces. Suppose $A_{n} \in \mathcal{B}(X, Y)$ for $n \in \mathbb {N}$ and $A x = \lim_{n \rightarrow \infty } A_{n} x$ exists for each $x$ in a dense subspace $S$ of $X$ .

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(a)

Show that if $\sup_{n}\left\| A_{n}\right\| < \infty$ then $A$ extends to a bounded map on $X$ , and $A x = \lim_{n \rightarrow \infty } A_{n} x$ for all $x \in X$ .

(b)

Give an example that shows that the hypothesis $\sup_{n}\left\| A_{n}\right\| < \infty$ in part (a) is necessary.

Problem 2.32

Prove Theorem 2.24 for the cases $p = 1$ and $p = \infty$ .

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Remark.

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Theorem 2.24: Fix $1 \leq p \leq \infty$ and any sequence of scalars $y = \left(y_{k}\right)$ . Then $\sum x_{k} y_{k}$ converges for all $x \in \ell^{p}$ if and only if $y \in \ell^{p^{\prime }}$ . Furthermore, in this case $T_{y} x = \left(x_{k} y_{k}\right)$ defines a bounded linear map of $\ell^{p}$ into $\ell^{1}$ , and

\sum _{k}\left|x_{k} y_{k}\right| = \left\| T_{y} x\right\| _{\ell ^{1}} \leq \left\| x\right\| _{\ell ^{p}}\left\| y\right\| _{\ell ^{p^{\prime }}}, \quad x \in \ell ^{p}.

Here $p^{\prime }$ is the dual index, $\frac{1}{p}+\frac{1}{p^{\prime }} = 1$ .

Problem 2.33

(a)(b)

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(a)

Let $X$ be a Banach space. Show that $S \subseteq X^{*}$ is bounded if and only if $\sup \left\{ \left| \left\langle x, x^{*} \right\rangle \right|: x^{*} \in S\right\} < \infty$ for each $x \in X$ .

(b)

Let $X$ be a normed linear space. Show that $S \subseteq X$ is bounded if and only if $\sup \left\{ \left| \left\langle x, x^{*} \right\rangle \right|: x \in S\right\} < \infty$ for each $x^{*} \in X^{*}$ .

Problem 2.34

(a)(b)

Fix $1 \leq p, q \leq \infty$ . Let $A = \left[a_{i j}\right]_{i, j \in \mathbb {N}}$ be an infinite matrix and set $a_{i} = \left(a_{i j}\right)_{j \in \mathbb {N}}$ for each $i \in \mathbb {N}$ . Suppose that

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(a)

$(A x)_{i} = \left\langle x, a_{i} \right\rangle = \sum_{j} a_{i j} x_{j}$ converges for each $x \in \ell^{p}$ and $i \in \mathbb {N}$ , and

(b)

$A x = \left((A x)_{i}\right)_{i \in \mathbb {N}} = \left( \left\langle x, a_{i} \right\rangle \right)_{i \in \mathbb {N}} \in \ell^{q}$ for each $x \in \ell^{p}$ .

Identifying the matrix $A$ with the map $x \mapsto A x$ , prove that $A \in \mathcal{B}\left(\ell^{p}, \ell^{q}\right)$ .

Problem 2.35

Let $X$ and $Y$ be Banach spaces. Show that $A \in \mathcal{B}(X, Y)$ is surjective if and only if $\operatorname {range}(A)$ is not meager in $Y$ .

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Problem 2.36

Show that if $T: X \rightarrow Y$ is a topological isomorphism of a normed space $X$ onto a normed space $Y$ , then a sequence $\left\{ x_{n}\right\}$ is complete in $X$ if and only if $\left\{ T x_{n}\right\}$ is complete in $Y$ .

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Problem 2.37

Let $X$ and $Y$ be normed linear spaces. Show that if $T: X \rightarrow Y$ is a topological isomorphism, then $\left\| T^{-1}\right\|^{-1}\left\| x\right\| \leq \left\| T x\right\| \leq \left\| T\right\| \left\| x\right\|$ for all $x \in X$ .

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Problem 2.38

(a)(b)

Let $X$ be a Banach space and $Y$ a normed linear space. Suppose that $L: X \rightarrow Y$ is bounded and linear. Prove that the following two statements are equivalent.

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(a)

There exists $c > 0$ such that $\left\| L x\right\| \geq c\left\| x\right\|$ for all $x \in X$ .

(b)

$L$ is injective and $\operatorname {range}(L)$ is closed.

Show further that, in case these hold, $L: X \rightarrow \operatorname {range}(L)$ is a topological isomorphism.

Problem 2.39

(a)(b)(c)

Given a sequence of scalars $\lambda = \left(\lambda_{k}\right)$ , define a mapping $T_{\lambda }$ on sequences $x = \left(x_{k}\right)$ by $T_{\lambda } x = \left(\lambda_{k} x_{k}\right)$ . Prove the following statements.

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(a)

$T_{\lambda }$ is a bounded map of $\ell^{2}$ into itself if and only if $\lambda \in \ell^{\infty }$ .

(b)

$T_{\lambda }$ is a topological isomorphism of $\ell^{2}$ onto itself if and only if $0 < \inf \left|\lambda_{k}\right| \leq \sup \left|\lambda_{k}\right| < \infty$ .

(c)

$T_{\lambda }$ is an isometric isomorphism of $\ell^{2}$ onto itself if and only if $\left|\lambda_{k}\right| = 1$ for every $n$ .

Problem 2.40

Let $X$ be a Banach space. Given $T \in \mathcal{B}(X)$ , define $T^{0} = I$ . Show that if $\left\| T\right\| < 1$ , then $I-T$ is a topological isomorphism of $X$ onto itself and $(I-T)^{-1} = \sum_{n = 0}^{\infty } T^{n}$ , where the series converges in operator norm (this is called a Neumann series for $(I-T)^{-1}$ ).

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Problem 2.41

Show that if $X$ is a Banach space, $Y$ is a normed linear space, and $T: X \rightarrow Y$ is a topological isomorphism, then $Y$ is a Banach space.

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Problem 2.42

(a)(b)

Prove Theorem 2.31.

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(a)

If $T: X \rightarrow Y$ is a topological isomorphism, then its adjoint $T^{*}: Y^{*} \rightarrow X^{*}$ is a topological isomorphism.

(b)

If $T$ is an isometric isomorphism, then so is $T^{*}$ .

Problem 2.43

Show that the operator $A^{\dagger } = B^{-1} P$ defined in Theorem 2.33 satisfies statements (a)-(c) of that theorem.

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Remark.

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Theorem 2.33: Let $H$ and $K$ be Hilbert spaces. Assume that $A \in \mathcal{B}(H, K)$ has closed range, and let $P$ be the orthogonal projection of $K$ onto $\operatorname {range}(A)$ . Then the mapping $B: \operatorname {ker}(A)^{\perp } \rightarrow \operatorname {range}(A)$ defined by $B x = A x$ for $x \in \operatorname {ker}(A)^{\perp }$ is a topological isomorphism, and $A^{\dagger } = B^{-1} P \in \mathcal{B}(K, H)$ satisfies the following:

$A A^{\dagger } y = y$ for every $y \in \operatorname {range}(A)$ ,
$A A^{\dagger }$ is the orthogonal projection of $K$ onto $\operatorname {range}(A)$ , and
$A^{\dagger } A$ is the orthogonal projection of $H$ onto $\operatorname {range}\left(A^{*}\right)$ .

Problem 2.44

(a)(b)(c)(d)

Assume that $A \in \mathcal{B}(H, K)$ has closed range, and let $A^{\dagger }$ be its pseudoinverse. Prove the following statements.

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(a)

$\operatorname {ker}\left(A^{\dagger }\right) = \operatorname {range}(A)^{\perp }$ .

(b)

$\operatorname {range}\left(A^{\dagger }\right) = \operatorname {ker}(A)^{\perp }$ .

(c)

$A A^{\dagger } y = y$ for all $y \in \operatorname {range}(A)$ .

(d)

$A^{\dagger }$ is the unique operator in $\mathcal{B}(K, H)$ that satisfies statements (a)-(c) above.

Problem 2.45

(a)(b)(c)

Let $H$ be a Hilbert space. Given a positive definite operator $A \in \mathcal{B}(H)$ , prove the following statements.

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(a)

$A$ is injective and has dense range.

(b)

$A$ is a topological isomorphism of $H$ onto itself if and only if it is surjective. Show by example that a positive definite operator need not be a topological isomorphism.

(c)

If $A$ is a surjective positive definite operator, then $(x, y) = \left\langle A x, y \right\rangle$ defines an inner product that is equivalent to the original inner product $\left\langle \cdot , \cdot \right\rangle$ on $H$ .

Problem 2.46

Prove the claim in Theorem 2.35 that $|||\cdot |||$ is a norm on $X$ and $X$ is complete with respect to this norm.

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Remark.

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Theorem 2.35 (Closed Graph Theorem): Let $X$ and $Y$ be Banach spaces. If $T: X \rightarrow Y$ is linear, then the following statements are equivalent.

$T$ is continuous.
If $x_{n} \rightarrow x$ in $X$ and $T x_{n} \rightarrow y$ in $Y$ , then $y = T x$ . In the proof of (b) $\Rightarrow$ (a) one assumes that statement (b) holds and defines $|||x||| = \left\| x\right\|_{X}+\left\| T x\right\|_{Y}$ , $x \in X$ .

Problem 2.47

Use the Closed Graph Theorem to give another proof of Theorem 2.24.

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Problem 2.48

(a)(b)

Use the Closed Graph Theorem to give another proof of Exercise 2.34.

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(a)

Using Theorem 2.24, show that $a_{i} \in \ell^{p^{\prime }}$ for each $i$ .

(b)

Using the Closed Graph Theorem, show that $A$ is a bounded map of $\ell^{p}$ into $\ell^{q}$ .

Problem 2.49

Let $C_{b}(\mathbb {R})$ and $C_{b}^{1}(\mathbb {R})$ be as in Exercise 1.22, and assume that the norm on both of these spaces is the uniform norm. In this case $C_{b}(\mathbb {R})$ is complete, but $C_{b}^{1}(\mathbb {R})$ is not. Show that the differentiation operator $D: C_{b}^{1}(\mathbb {R}) \rightarrow C_{b}(\mathbb {R})$ given by $D f = f^{\prime }$ is unbounded, but has a closed graph, i.e., if $f_{n} \rightarrow f$ uniformly and $f_{n}^{\prime } \rightarrow g$ uniformly then $f^{\prime } = g$ .

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Problem 2.50

Prove part (b) of Theorem 2.38.

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Problem 2.51

(a)(b)

In this exercise we will denote the components of $x \in \ell^{p}$ by $x = (x(k))$ .

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(a)

Given $1 < p < \infty$ and $x_{n}, y \in \ell^{p}$ , show that $x_{n} \xrightarrow {\mathrm{w}} y$ in $\ell^{p}$ if and only if $\sup \left\| x_{n}\right\|_{\ell^{p}} < \infty$ and $x_{n}$ converges componentwise to $y$ , i.e., $\lim_{n \rightarrow \infty } x_{n}(k) = y(k)$ for each $k \in \mathbb {N}$ . Does either implication remain valid if $p = 1$ ?

(b)

Given $1 \leq p \leq \infty$ and $x_{n}, y \in \ell^{p}$ , show that $x_{n} \xrightarrow {\mathrm{w}^{*}} y$ in $\ell^{p}$ if and only if $x_{n}$ converges componentwise to $y$ and $\sup \left\| x_{n}\right\|_{\ell^{p}} < \infty$ (recall that $\ell^{1} \cong c_{0}{ }^{*}$ and $\ell^{p^{\prime }} \cong \left(\ell^{p}\right)^{*}$ for $1 \leq p < \infty$ .

Problem 2.52

Show that if $\left\{ x_{n}\right\}$ is an orthonormal sequence in a Hilbert space $H$ , then $x_{n} \xrightarrow {\mathrm{w}} 0$ .

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