Math Problems Vault

Problem 4.1

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

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

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Problem 4.2

(a)(b)(c)

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

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(a)

Any Hamel basis for $X$ must be uncountable.

(b)

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

(c)

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

Problem 4.3

Let $\left\{ x_{i}\right\}_{i \in I}$ be a Hamel basis for an infinite-dimensional Banach space $X$ . Then each $x \in X$ can be written uniquely as $x = \sum_{i \in I} a_{i}(x) x_{i}$ where at most finitely many of the scalars $a_{i}(x)$ are nonzero. Each $a_{i}$ is a linear functional on $X$ , and we call $\left\{ a_{i}\right\}_{i \in I}$ the sequence of coefficient functionals associated with the Hamel basis $\left\{ x_{i}\right\}_{i \in I}$ . 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 $a_{i}$ to be continuous.

(b)

Show that $a_{i}\left(x_{j}\right) = \delta_{i j}$ for $i, j \in I$ , where $\delta_{i j}$ is the Kronecker delta.

(c)

Let $J = \left\{ i \in I: a_{i} \text{ is continuous}\right\}$ . Show that $\sup_{j \in J}\left\| a_{j}\right\| < \infty$ .

(d)

Show that at most finitely many functionals $a_{i}$ can be continuous, i.e., $J$ is finite.

(e)

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

Problem 4.4

(a)(b)

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

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(a)

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

(b)

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

Problem 4.5

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

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Problem 4.6

(a)(b)(c)

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

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(a)

Show that $\left\{ \lambda_{n} x_{n}\right\}$ is a basis for $X$ . In particular, $\left\{ x_{n} /\left\| x_{n}\right\| \right\}$ is a normalized basis for $X$ .

(b)

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

(c)

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

Problem 4.7

(a)(b)(c)

Let $H$ be a separable Hilbert space.

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(a)

Show that an orthonormal basis for $H$ is a normalized unconditional basis for $H$ .

(b)

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

(c)

Give an example of a basis $\left\{ x_{n}\right\}$ for $H$ that contains no orthogonal subsequences (equivalently, $\left\langle x_{m}, x_{n} \right\rangle \neq 0$ for all $m \neq n$ ).

Problem 4.8

(a)(b)(c)

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(a)

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

(b)

By Exercise 1.20, $c = \left\{ x = \left(x_{n}\right) \in \ell^{\infty }: \lim_{n \rightarrow \infty } x_{n} \text{ exists}\right\}$ is a closed subspace of $\ell^{\infty }$ , and $c_{0}$ is a proper closed subspace of $c$ . Find a vector $\delta_{0} \in c$ such that $\left\{ \delta_{n}\right\}_{n \geq 0}$ is a normalized unconditional basis for $c$ .

(c)

Show that $c^{*}$ is isometrically isomorphic to $\ell^{1}$ . Compare Exercise 1.75, which shows that we also have $c_{0}{ }^{*} \cong \ell^{1}$ , and Exercise 4.22, which shows that $c$ and $c_{0}$ are topologically isomorphic but not isometrically isomorphic.

Problem 4.9

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

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Problem 4.10

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

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Problem 4.11

Show that the standard basis $\left\{ \delta_{n}\right\}_{n \in \mathbb {N}}$ is a monotone basis for $\ell^{p}$ for each index $1 \leq p < \infty$ , and is also a monotone basis for $c_{0}$ . Is the basis $\left\{ \delta_{n}\right\}_{n \geq 0}$ for $c$ 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 $\left\{ x_{n}\right\}$ is a Schauder basis for a Hilbert space $H$ , and its biorthogonal system is $\left\{ y_{n}\right\}$ . Show that if $\left\| x_{n}\right\| = \left\| y_{n}\right\| = 1$ for every $n$ , then $\left\{ x_{n}\right\}$ is an orthonormal basis for $H$ .

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Problem 4.14

(a)(b)

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(a)

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

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 $\ell^{1}$ onto $X$ .

(b)

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

Problem 4.15

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

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(a)

Show that $|||\cdot |||$ is a norm on $X$ , and $\left\| x\right\| \leq \left\| x\right\|$ for all $x \in X$ .

(b)

Suppose that $\left\{ y_{n}\right\}$ is a Cauchy sequence in $X$ with respect to $|||\cdot |||$ . With $N$ fixed, show that $\left\{ S_{N} y_{n}\right\}_{n \in \mathbb {N}}$ is Cauchy with respect to $\left\| \cdot \right\|$ . Let $z_{N}$ be such that $\left\| z_{N}-S_{N} y_{n}\right\| \rightarrow 0$ as $n \rightarrow \infty$ , and observe that $z_{N} \in \operatorname {span}\left\{ x_{1}, \ldots , x_{N}\right\}$ .

(c)

Show that for each $N \in \mathbb {N}$ we have $\lim_{n \rightarrow \infty }\left(\sup_{N}\left\| z_{N}-S_{N} y_{n}\right\| \right) = 0$ , and use this to show that $\left\{ z_{N}\right\}_{N \in \mathbb {N}}$ is Cauchy with respect to $\left\| \cdot \right\|$ . Let $y \in X$ be the element such that $\left\| y-z_{N}\right\| \rightarrow 0$ .

(d)

Show that $S_{N}\left(z_{N+1}\right) = z_{N}$ , and use this to show that $z_{N} = \sum_{n = 1}^{N} c_{n} x_{n}$ where $c_{n}$ is independent of $N$ .

(e)

Show that $y = \sum_{n = 1}^{\infty } c_{n} x_{n}$ , and hence $z_{N} = S_{N} y$ . Use this to show that $\left\| y-y_{n}\right\| \rightarrow 0$ , and conclude that $X$ is complete with respect to $|||\cdot |||$ .

(f)

Show that $\left\| \cdot \right\|$ and $|||\cdot |||$ are equivalent norms on $X$ , and use this to show that $\mathcal{C} = \sup_{N}\left\| S_{N}\right\| < \infty$ . Conclude that $\left\{ x_{n}\right\}$ is a Schauder basis for $X$ .

Problem 4.16

(a)(b)

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

\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 $X$ has a basis, then $X$ has the approximation property.

(b)

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

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

Problem 4.17

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

Let $X$ be a Banach space.

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(a)

Show that if $\left\{ x_{n}\right\}$ is a basis for $X$ , then the only possible vector $y \in X$ such that $x_{n} \xrightarrow {\mathrm{w}} y$ is $y = 0$ (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 $\left\{ x_{n}\right\}$ is a basic sequence in $X$ .

(c)

Give an example of a basis $\left\{ x_{n}\right\}$ for a Banach space $X$ such that $x_{n} \xrightarrow {\mathrm{w}} 0$ .

(d)

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

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 $X$ and $Y$ be Banach spaces. If $\left\{ x_{n}\right\}$ is a basis for $X$ and $\left\{ y_{n}\right\}$ is a basis for $Y$ , then the following two statements are equivalent.

$\left\{ x_{n}\right\}$ is equivalent to $\left\{ y_{n}\right\}$ .
$\sum c_{n} x_{n}$ converges in $X$ if and only if $\sum c_{n} y_{n}$ converges in $Y$ . Definition 4.19: A basis $\left\{ x_{n}\right\}$ for $X$ is equivalent to a basis $\left\{ y_{n}\right\}$ for $Y$ if there exists a topological isomorphism $T: X \rightarrow Y$ such that $T x_{n} = y_{n}$ for all $n$ .

Problem 4.19

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

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Problem 4.20

(a)(b)(c)

Suppose $\left\{ x_{n}\right\}$ is a basis for a Banach space $X$ that is equivalent to a basis $\left\{ y_{n}\right\}$ for a Banach space $Y$ .

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(a)

Show that $\left\{ x_{n}\right\}$ is a bounded basis for $X$ if and only if $\left\{ y_{n}\right\}$ is a bounded basis for $Y$ .

(b)

Show that $\left\{ x_{n}\right\}$ is an unconditional basis for $X$ if and only if $\left\{ y_{n}\right\}$ is an unconditional basis for $Y$ .

(c)

Show that $\left\{ x_{n}\right\}$ is an absolutely convergent basis for $X$ if and only if $\left\{ y_{n}\right\}$ is an absolutely convergent basis for $Y$ .

Problem 4.21

Let $\left\{ x_{n}\right\}$ be a basis for a Banach space $X$ and $\left\{ y_{n}\right\}$ a basis for a Banach space $Y$ . Show that $\left\{ x_{n}\right\}$ is equivalent to $\left\{ y_{n}\right\}$ if and only if there exist constants $C_{1}, C_{2} > 0$ such that for all $N \in \mathbb {N}$ and $c_{1}, \ldots , c_{N} \in \mathbf{F}$ we have

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\| .

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Problem 4.22

(a)(b)(c)

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(a)

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

(b)

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

(c)

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

Problem 4.23

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

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Problem 4.24

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

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(a)

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

(b)

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

(c)

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

(d)

Show that if $1 \leq p < q \leq \infty$ , then $L^{q}(\mathbf{T}) \subsetneq L^{p}(\mathbf{T})$ .

Problem 4.25

Prove Theorem 4.27.

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Remark.

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

Problem 4.26

(a)(b)

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

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(a)

$Y$ is a Banach space.

(b)

If $\left\{ x_{n}\right\}$ is a weak basis for $X$ , then $Y$ is topologically isomorphic to $X$ via $\left(c_{n}\right) \mapsto \sum c_{n} x_{n}$ .

Problem 4.27

Finish the details of Example 4.31.

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Remark.

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

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

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

Problem 4.28

(a)(b)(c)

Prove the statements made in Example 4.32:

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(a)

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

(b)

The representation $x = \sum \left\langle x, y_{n} \right\rangle x_{n}$ is the unique representation of $x$ with respect to weak* convergence, i.e., $\left\{ x_{n}\right\}$ is a weak* basis for $\ell^{1}$ .

(c)

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

Problem 4.29

(a)(b)

Prove the statements made in Example 4.33:

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(a)

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

(b)

$\sum x_{n} = 0$ with weak* convergence, so $\left\{ x_{n}\right\}$ is not weak* $\omega$ -independent; hence $x = \sum \left\langle x, y_{n} \right\rangle x_{n}$ is not the unique weak*-convergent representation of $x$ , so $\left\{ x_{n}\right\}$ is not a weak* basis for $\ell^{1}$ .