Part I

A Primer on Functional Analysis

[132/99%]

Show

LaTeX

Show LaTeX

On this page

1 · Banach Spaces and Operator Theory

2 · Functional Analysis

Chapter 1Banach Spaces and Operator Theory[80/99%]

Problem 1.1

Show that if $\left\| \cdot \right\|$ is a norm on the scalar field $\mathbf{F}$ , then there exists a positive number $\lambda > 0$ such that $\left\| x\right\| = \lambda \left|x\right|$ , where $\left|x\right|$ is the absolute value of $x$ .

Origin: Stated according to the original source

Problem 1.2

(a)(b)(c)(d)(e)(f)(g)

Given a normed linear space $X$ , prove the following facts.

Origin: Stated according to the original source

(a)

Every convergent sequence in $X$ is Cauchy, and the limit of a convergent sequence is unique.

(b)

Every Cauchy sequence in $X$ is bounded.

(c)

(d)

Continuity of the norm: $x_{n} \rightarrow x \Longrightarrow \left\| x_{n}\right\| \rightarrow \left\| x\right\|$ .

(e)

Continuity of vector addition:

x_{n} \rightarrow x \text{ and } y_{n} \rightarrow y \quad \Longrightarrow \quad x_{n}+y_{n} \rightarrow x+y

(f)

Continuity of scalar multiplication:

x_{n} \rightarrow x \text{ and } c_{n} \rightarrow c \Longrightarrow c_{n} x_{n} \rightarrow c x

(g)

Convexity of open balls: If $x, y \in B_{r}(z)$ then

\theta x+(1-\theta ) y \in B_{r}(z) \text{ for all } 0 \leq \theta \leq 1

Problem 1.3

(a)(b)

Origin: Stated according to the original source

(a)

Assuming that the functions $|\cdot |_{p}$ on $\mathbf{F}^{d}$ given in equation (1.1) are norms on $\mathbf{F}^{d}$ , show that any two of these norms are equivalent.

(b)

Assuming that $\mathbf{F}$ is complete with respect to absolute value, show that $\mathbf{F}^{d}$ is complete with respect to any one of the norms $|\cdot |_{p}$ .

Problem 1.4

Let $V$ be a finite-dimensional vector space. Prove that the function $\left\| \cdot \right\|_{1}$ on $V$ defined in Example 1.6 is a norm, and that $V$ is complete with respect to this norm.

Origin: Stated according to the original source

Problem 1.5

Show that if vectors $x_{n}$ in a normed space $X$ satisfy $\left\| x_{n+1}-x_{n}\right\| < 2^{-n}$ for every $n \in \mathbb {N}$ , then $\left\{ x_{n}\right\}$ is a Cauchy sequence in $X$ .

Origin: Stated according to the original source

Problem 1.6

Let $\left\{ x_{n}\right\}$ be a sequence in a normed space $X$ , and let $x \in X$ be fixed. Suppose that every subsequence $\left\{ y_{n}\right\}$ of $\left\{ x_{n}\right\}$ has a subsequence $\left\{ z_{n}\right\}$ of $\left\{ y_{n}\right\}$ such that $z_{n} \rightarrow x$ . Show that $x_{n} \rightarrow x$ .

Origin: Stated according to the original source

Problem 1.7

Let $X$ be a complex Banach space. Let $X_{\mathbb {R}} = X$ as a set, but consider $X_{\mathbb {R}}$ as a vector space over the real field. That is, vector addition in $X_{\mathbb {R}}$ is defined just as in $X$ , but scalar multiplication in $X_{\mathbb {R}}$ is restricted to multiplication by real scalars. Let $\left\| \cdot \right\|_{\mathbb {R}} = \left\| \cdot \right\|$ , and show that $(X_{\mathbb {R}},\left\| \cdot \right\|_{\mathbb {R}})$ is a real Banach space.

Origin: Stated according to the original source

Problem 1.8

(a)(b)(c)

We say that a set $X$ (not necessarily a vector space) is a metric space if for each $x, y \in X$ there exists a real number $\mathrm{d}(x, y)$ such that for all $x, y$ , $z \in X$ we have: i. i. $\mathrm{d}(x, y) \geq 0$ , ii. $\mathrm{d}(x, y) = 0$ if and only if $x = y$ , iii. $\mathrm{d}(x, y) = \mathrm{d}(y, x)$ , and iv. the Triangle Inequality: $\mathrm{d}(x, y) \leq \mathrm{d}(x, z)+\mathrm{d}(y, z)$ .

In this case we call d a metric on $X$ , and we refer to $\mathrm{d}(x, y)$ as the distance between $x$ and $y$ .

Origin: Stated according to the original source

(a)

Show that if $X$ is a normed space, then $\mathrm{d}(x, y) = \left\| x-y\right\|$ is a metric on $X$ .

(b)

Make a definition of convergent and Cauchy sequences in a metric space, and show that every convergent sequence is Cauchy.

(c)

Define a metric space to be complete if every Cauchy sequence in $X$ converges to an element of $X$ . Let $X = \mathbb {Q}$ and set $\mathrm{d}(x, y) = \left|x-y\right|$ , the ordinary absolute value of the difference of $x$ and $y$ . Show that d is a metric on $\mathbb {Q}$ , but $\mathbb {Q}$ is incomplete with respect to this metric.

Problem 1.9

(a)(b)(c)

Fix $0 < p < 1$ , and define $\ell^{p}$ by equation (1.2) and $\left\| \cdot \right\|_{\ell^{p}}$ by equation (1.3).

Origin: Stated according to the original source

(a)

Show that $\left\| \cdot \right\|_{\ell^{p}}$ fails the Triangle Inequality and hence is not a norm on $\ell^{p}$ .

(b)

Show that $\left\| x+y\right\|_{\ell^{p}}^{p} \leq \left\| x\right\|_{\ell^{p}}^{p}+\left\| y\right\|_{\ell^{p}}^{p}$ , and use this to show that $\ell^{p}$ is a vector space and $\mathrm{d}(x, y) = \left\| x-y\right\|_{\ell^{p}}^{p}$ is a metric on $\ell^{p}$ .

(c)

Let $B = \left\{ x \in \ell^{p}: \mathrm{d}(x, 0) < 1\right\}$ be the "open unit ball" with respect to the metric d, and show that $B$ is not convex. Use this to show that there is no norm $\left\| \cdot \right\|$ on $\ell^{p}$ such that $\mathrm{d}(x, y) = \left\| x-y\right\|$ .

Problem 1.10

(a)(b)

Origin: Stated according to the original source

(a)

Show that if $0 < \theta < 1$ , then $t^{\theta } \leq \theta t+(1-\theta )$ for $t \geq 0$ , with equality if and only if $t = 1$ .

(b)

Suppose that $1 < p < \infty$ and $a, b \geq 0$ . Apply part (a) with $t = a^{p} b^{-p^{\prime }}$ and $\theta = 1 / p$ to show that $a b \leq a^{p} / p+b^{p^{\prime }} / p^{\prime }$ , with equality if and only if $b = a^{p-1}$ .

Problem 1.11

Show that equality holds in Hölder's Inequality for sequences (part (a) of Theorem 1.12) if and only if there exist scalars $\alpha , \beta$ , not both zero, such that $\alpha \left|x_{k}\right|^{p} = \beta \left|y_{k}\right|^{p^{\prime }}$ for each $k \in I$ .

Origin: Stated according to the original source

Problem 1.12

Show that if $1 \leq p < q \leq \infty$ , then $\ell^{p} \subsetneq \ell^{q}$ and $\left\| x\right\|_{\ell^{q}} \leq \left\| x\right\|_{\ell^{p}}$ for all $x \in \ell^{p}$ .

Origin: Stated according to the original source

Problem 1.13

Let $E \subseteq \mathbb {R}$ be measurable with $\left|E\right| < \infty$ . Show that if $1 \leq p < q \leq \infty$ , then $L^{q}(E) \subsetneq L^{p}(E)$ and $\left\| f\right\|_{L^{p}} \leq \left|E\right|^{\frac{1}{p}-\frac{1}{q}}\left\| f\right\|_{L^{q}}$ for all $f \in L^{p}(E)$ .

Origin: Stated according to the original source

Problem 1.14

Show that if $x \in \ell^{q}$ for some finite $q$ , then $\left\| x\right\|_{\ell^{p}} \rightarrow \left\| x\right\|_{\ell^{\infty }}$ as $p \rightarrow \infty$ , but this can fail if $x \notin \ell^{q}$ for any finite $q$ .

Origin: Stated according to the original source

Problem 1.15

Given an arbitrary index set $I$ , define $\ell^{\infty }(I)$ to be the space of all bounded sequences $x = \left(x_{i}\right)_{i \in I}$ indexed by $I$ , with $\left\| x\right\|_{\infty } = \sup_{i \in I}\left|x_{i}\right|$ . For $1 \leq p < \infty$ let $\ell^{p}(I)$ consist of all sequences $x = \left(x_{i}\right)_{i \in I}$ with at most countably many nonzero components such that $\left\| x\right\|_{\ell^{p}}^{p} = \sum \left|x_{i}\right|^{p} < \infty$ . Show that each of these spaces $\ell^{p}(I)$ is a Banach space with respect to $\left\| \cdot \right\|_{\ell^{p}}$ . What is $\ell^{p}(I)$ if $I = \left\{ 1, \ldots , d\right\}$ ?

Origin: Stated according to the original source

Problem 1.16

(a)(b)

Let $X, Y$ be normed linear spaces. Given $1 \leq p < \infty , x \in X$ , and $y \in Y$ , define $\left\| (x, y)\right\|_{p} = \left(\left\| x\right\|_{X}^{p}+\left\| y\right\|_{Y}^{p}\right)^{1 / p}$ and $\left\| (x, y)\right\|_{\infty } = \max \left\{ \left\| x\right\|_{X},\left\| y\right\|_{Y}\right\}$ .

Origin: Stated according to the original source

(a)

Prove that $\left\| \cdot \right\|_{p}$ is a norm on the Cartesian product $X \times Y$ , and $\left\| \cdot \right\|_{p}$ and $\left\| \cdot \right\|_{q}$ are equivalent norms on $X \times Y$ for any $1 \leq p, q \leq \infty$ .

(b)

Show that if $X$ and $Y$ are Banach spaces, then $X \times Y$ is a Banach space with respect to $\left\| \cdot \right\|_{p}$ .

Problem 1.17

(a)(b)(c)

Prove Lemma 1.16: Let $E$ be a subset of a metric space $X$ .

Origin: Stated according to the original source

(a)

$E$ is closed if and only if $E$ contains all of its limit points.

(b)

$\bar{E} = E \cup \left\{ x \in X: x\text{ is a limit point of }E\right\}$ . Consequently, $E$ is closed if and only if $E = \bar{E}$ .

(c)

$E$ is dense in $X$ if and only if every $x \in X$ is a limit point of $E$ .

Problem 1.18

Prove Lemma 1.17.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

Lemma 1.17: Let $S$ be a subspace of a Banach space $X$ . Then $S$ is a Banach space with respect to the norm on $X$ if and only if $S$ is a closed subset of $X$ .

Problem 1.19

Show that if $f \in C_{b}(\mathbb {R})$ , then $\operatorname {ess}_{\sup }^{t \in \mathbb {R}}\left|f(t)\right| = \sup_{t \in \mathbb {R}}\left|f(t)\right|$ , and consequently $\left\| f\right\|_{L^{\infty }} = \left\| f\right\|_{\infty }$ .

Origin: Stated according to the original source

Problem 1.20

(a)(b)(c)

Origin: Stated according to the original source

(a)

Show that $c$ and $c_{0}$ are closed subspaces of $\ell^{\infty }$ .

(b)

Show that $c_{00}$ is a proper, dense subspace of $c_{0}$ , and hence is not closed with respect to the norm $\left\| \cdot \right\|_{\ell \infty }$ .

(c)

Let $\left\{ \delta_{n}\right\}$ denote the sequence of standard basis vectors (see Example 1.30). Given $x = \left(x_{n}\right) \in c_{0}$ , show that $x = \sum x_{n} \delta_{n}$ , where the series converges with respect to the norm $\left\| \cdot \right\|_{\ell \infty }$ . Show further that the scalars $x_{n}$ in this representation are unique.

Problem 1.21

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

Origin: Stated according to the original source

(a)

Show that $C_{b}(\mathbb {R})$ is a Banach space with respect to the uniform norm $\left\| \cdot \right\|_{\infty }$ .

(b)

Show that $C_{0}(\mathbb {R})$ is a closed subspace of $C_{b}(\mathbb {R})$ .

(c)

Show that $C_{c}(\mathbb {R})$ is a proper, dense subspace of $C_{0}(\mathbb {R})$ , and hence is not closed with respect to $\left\| \cdot \right\|_{\infty }$ .

(d)

Let $C(\mathbf{T})$ be the set of all continuous functions $f \in C(\mathbb {R})$ that are 1 -periodic, i.e., $f(t+1) = f(t)$ for every $t \in \mathbb {R}$ . Show that $C(\mathbf{T})$ is a closed subspace of $C_{b}(\mathbb {R})$ .

Problem 1.22

(a)(b)

Let $C_{b}^{m}(\mathbb {R})$ be the space of all $m$ -times differentiable functions on $\mathbb {R}$ each of whose derivatives is bounded and continuous, i.e.,

C_{b}^{m}(\mathbb {R}) = \left\{ f \in C_{b}(\mathbb {R}): f, f^{\prime }, \ldots , f^{(m)} \in C_{b}(\mathbb {R})\right\} .

Origin: Stated according to the original source

(a)

Show that $C_{b}^{m}(\mathbb {R})$ is a Banach space with respect to the norm

\left\| f\right\| _{C_{b}^{m}} = \left\| f\right\| _{\infty }+\left\| f^{\prime }\right\| _{\infty }+\cdots +\left\| f^{(m)}\right\| _{\infty }

and

C_{0}^{m}(\mathbb {R}) = \left\{ f \in C_{0}(\mathbb {R}): f, f^{\prime }, \ldots , f^{(m)} \in C_{0}(\mathbb {R})\right\}

is a subspace of $C_{b}^{m}(\mathbb {R})$ that is also a Banach space with respect to the same norm.

(b)

Now we change the norm on $C_{b}^{1}(\mathbb {R})$ . Show that $(C_{b}^{1}(\mathbb {R}),\left\| \cdot \right\|_{\infty })$ is a normed space, but is not complete.

Problem 1.23

(a)(b)(c)

We say that a function $f: \mathbb {R} \rightarrow \mathbf{F}$ is Hölder continuous with exponent $\alpha > 0$ if there exists a constant $K > 0$ such that

\forall x, y \in \mathbb {R}, \quad \left|f(x)-f(y)\right| \leq K\left|x-y\right|^{\alpha }

A function that is Hölder continuous with exponent $\alpha = 1$ is said to be Lipschitz.

Origin: Stated according to the original source

(a)

Show that if $f$ is Hölder continuous for some $\alpha > 1$ , then $f$ is constant.

(b)

Show that if $f$ is differentiable on $\mathbb {R}$ and $f^{\prime }$ is bounded, then $f$ is Lipschitz. Find a function $g$ that is Lipschitz but is not differentiable at every point.

(c)

Given $0 < \alpha < 1$ , define

C^{\alpha }(\mathbb {R}) = \left\{ f \in C(\mathbb {R}): f \text{ is Hölder continuous with exponent } \alpha \right\} \text{. }

Show that $C^{\alpha }(\mathbb {R})$ is a Banach space with respect to the norm

\left\| f\right\| _{C^{\alpha }} = \left|f(0)\right|+\sup _{x \neq y} \frac{f(x)-f(y)}{\left|x-y\right|^{\alpha }}

Problem 1.24

(a)(b)(c)

Consider the two functions $\varphi_{1}, \varphi_{2}$ pictured in Figure 1.1. The function $\varphi_{1}$ takes the constant value $1 / 2$ on the interval $(1 / 3,2 / 3)$ that is removed in the first stage of the construction of the classical Cantor middle-thirds set, and is linear on the remaining intervals. The function $\varphi_{2}$ also takes the same constant $1 / 2$ on the interval $(1 / 3,2 / 3)$ but additionally is constant with values $1 / 4$ and $3 / 4$ on the two intervals $(1 / 9,2 / 9)$ and $(7 / 9,8 / 9)$ that are removed in the second stage of the construction of the Cantor set. Continue this process, defining $\varphi_{3}, \varphi_{4}, \ldots$ , and prove the following facts.

$Fig. 1.1. Top left: The function \varphi _{1}. Top right: The function \varphi _{2}. Bottom: The Devil’s staircase (Cantor-Lebesgue function).$ Fig. 1.1. Top left: The function \varphi _{1}. Top right: The function \varphi _{2}. Bottom: The Devil’s staircase (Cantor-Lebesgue function).

Origin: Stated according to the original source

(a)

Each $\varphi_{k}$ is monotone increasing on $[0,1]$ , and $\left|\varphi_{k+1}(t)-\varphi_{k}(t)\right| < 2^{-k}$ for every $t \in [0,1]$ .

(b)

$\varphi (t) = \lim_{k \rightarrow \infty } \varphi_{k}(t)$ converges uniformly on [ 0,1 ]. The limit function $\varphi$ is called the Cantor-Lebesgue function or, more picturesquely, the Devil's staircase.

(c)

The Cantor-Lebesgue function is Hölder continuous on the interval $[0,1]$ precisely for exponents $\alpha$ in the range $0 < \alpha \leq \log_{3} 2 \approx 0.6309 \ldots$ . In particular, $\varphi$ is continuous and monotone increasing on $[0,1]$ but it is not Lipschitz. Even so, $\varphi$ is differentiable for a.e. $t \in [0,1]$ , and $\varphi^{\prime }(t) = 0$ a.e.

Problem 1.25

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

Let $X$ be a normed linear space that is not complete. Let $\mathcal{C}$ be the set of all Cauchy sequences in $X$ , and define a relation $\sim$ on $\mathcal{C}$ by declaring that $\left\{ x_{n}\right\} \sim \left\{ y_{n}\right\}$ if $\lim_{n \rightarrow \infty }\left\| x_{n}-y_{n}\right\| = 0$ .

Origin: Stated according to the original source

(a)

Show that $\sim$ is an equivalence relation on $\mathcal{C}$ .

(b)

Let $\left[x_{n}\right] = \left\{ \left\{ y_{n}\right\} :\left\{ y_{n}\right\} \sim \left\{ x_{n}\right\} \right\}$ denote the equivalence class of $\left\{ x_{n}\right\}$ under the relation $\sim$ . Let $\widetilde{X}$ be the set of all equivalence classes $\left[x_{n}\right]$ . Define $\left\| \left[x_{n}\right]\right\|_{\widetilde{X}} = \lim_{n \rightarrow \infty }\left\| x_{n}\right\|$ . Prove that $\left\| \cdot \right\|_{\widetilde{X}}$ is a well-defined norm on $\widetilde{X}$ .

(c)

Given $x \in X$ , let $[x]$ denote the equivalence class of the Cauchy sequence $\left\{ x, x, x, \ldots \right\}$ . Show that $T: x \mapsto [x]$ is an isometric map of $X$ into $\widetilde{X}$ , where isometric means that $\left\| T x\right\|_{\widetilde{X}} = \left\| x\right\|$ for every $x$ (see Definition 1.60). Show also that $T(X)$ is a dense subspace of $\widetilde{X}$ (so, in the sense of identifying of $X$ with $T(X)$ , we can consider $X$ to be a subspace of $\widetilde{X}$ ).

(d)

Show that $\widetilde{X}$ is a Banach space with respect to $\left\| \cdot \right\|_{\widetilde{X}}$ . We call $\widetilde{X}$ the completion of $X$ .

(e)

Prove that $\widetilde{X}$ is unique in the sense that if $Y$ is a Banach space and $U: X \rightarrow Y$ is a linear isometry such that $U(X)$ is dense in $Y$ , then there exists a linear isometric bijection $V: Y \rightarrow \widetilde{X}$ .

Problem 1.26

Suppose that $\sum x_{n}$ and $\sum y_{n}$ are convergent series in a normed space $X$ . Show that $\sum \left(x_{n}+y_{n}\right)$ is convergent and equals $\sum x_{n}+\sum y_{n}$ .

Origin: Stated according to the original source

Problem 1.27

Show that $C_{0}(\mathbb {R})$ is separable.

Origin: Stated according to the original source

Problem 1.28

(a)(b)(c)

Origin: Stated according to the original source

(a)

Show that if $X$ is a normed linear space and there exists an uncountable set $S \subseteq X$ such that $\left\| x-y\right\| = 1$ for every $x \neq y \in S$ , then $X$ is not separable.

(b)

Let $D = \left\{ x = \left(x_{1}, x_{2}, \ldots \right) \in \ell^{\infty }: x_{k} = 0,1 \text{ for each } k\right\}$ . Show that $D$ is uncountable and that if $x, y$ are two distinct vectors in $D$ , then $\left\| x-y\right\|_{\ell \infty } = 1$ . Conclude that $\ell^{\infty }$ is not separable.

(c)

Show that $L^{\infty }(\mathbb {R})$ is not separable.

Problem 1.29

Let $\left(c_{k}\right)_{k \geq 0}$ be a fixed sequence of real numbers. Show that if the series $\sum_{k = 0}^{\infty } c_{k} y^{k}$ converges for some $y \in \mathbb {R}$ , then the series $f(x) = \sum_{k = 0}^{\infty } c_{k} x^{k}$ converges absolutely for all $\left|x\right| < \left|y\right|$ , and show that this function $f$ is infinitely differentiable for all $x$ with $\left|x\right| < \left|y\right|$ .

Origin: Stated according to the original source

Problem 1.30

(a)(b)(c)

An $n \times n$ matrix $A$ is said to be positive definite if $A x \cdot x > 0$ for all $x \in \mathbf{F}^{n}$ (compare Definition 2.14).

Origin: Stated according to the original source

(a)

Show that if $A$ is a positive definite $n \times n$ matrix then equation (1.13) defines an inner product on $\mathbf{F}^{d}$ .

(b)

Show that if $\left\langle \cdot , \cdot \right\rangle$ is an inner product on $\mathbf{F}^{d}$ then there exists some positive definite matrix $A$ such that equation (1.13) holds.

(c)

Prove that the inner products on $\mathbf{F}^{d}$ defined by equation (1.13) are all equivalent.

Problem 1.31

(a)(b)(c)

Prove Lemma 1.36.

Origin: Stated according to the original source

(a)

Polar Identity: $\left\| f+g\right\|^{2} = \left\| f\right\|^{2}+2 \operatorname {Re} \left\langle f, g \right\rangle +\left\| g\right\|^{2}$ .

(b)

Pythagorean Theorem: if $f \perp g$ then $\left\| f \pm g\right\|^{2} = \left\| f\right\|^{2}+\left\| g\right\|^{2}$ .

(c)

Parallelogram Law: $\left\| f+g\right\|^{2}+\left\| f-g\right\|^{2} = 2\left\| f\right\|^{2}+2\left\| g\right\|^{2}$ .

Problem 1.32

Show that if $p \neq 2$ then the norm $\left\| \cdot \right\|_{\ell^{p}}$ on $\ell^{p}$ is not induced from any inner product on $\ell^{p}$ .

Origin: Stated according to the original source

Problem 1.33

Show that equality holds in the Cauchy-Bunyakovski-Schwarz Inequality if and only if $x = c y$ or $y = c x$ for some scalar $c \in \mathbf{F}$ .

Origin: Stated according to the original source

Problem 1.34

We say that $\left\langle \cdot , \cdot \right\rangle$ is a semi-inner product on a vector space $H$ if properties (a), (c), and (d) of Definition 1.33 are satisfied. Prove that if $\left\langle \cdot , \cdot \right\rangle$ is a semi-inner product then $\left\| x\right\| = \left\langle x, x \right\rangle^{1 / 2}$ defines a seminorm on $H$ , and $\left| \left\langle x, y \right\rangle \right| \leq \left\| x\right\| \left\| y\right\|$ for all $x, y \in H$ .

Origin: Stated according to the original source

Problem 1.35

Given a closed subspace $M$ of a Hilbert space $H$ and given $x \in H$ , prove that the point in $M$ closest to $x$ is unique.

Origin: Stated according to the original source

Problem 1.36

Prove Lemma 1.42.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

Lemma 1.42: If $A$ is a subset of a Hilbert space $H$ , then $A^{\perp }$ is a closed subspace of $H$ .

Problem 1.37

Prove Theorem 1.43.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

Theorem 1.43: If $M$ is a closed subspace of a Hilbert space $H$ and $x \in H$ , then the following statements are equivalent: (a) $x = p+e$ where $p$ is the orthogonal projection of $x$ onto $M$ ; (b) $x = p+e$ where $p \in M$ and $e \in M^{\perp }$ ; (c) $x = p+e$ where $e$ is the orthogonal projection of $x$ onto $M^{\perp }$ .

Problem 1.38

(a)(b)(c)

Prove Lemma 1.44.

Origin: Stated according to the original source

(a)

If $M$ is a closed subspace of $H$ , then $\left(M^{\perp }\right)^{\perp } = M$ .

(b)

If $A$ is any subset of $H$ , then $A^{\perp } = \left(\operatorname {span}(A)\right)^{\perp } = \left(\overline{\operatorname {span}}(A)\right)^{\perp }$ , and $\left(A^{\perp }\right)^{\perp } = \overline{\operatorname {span}}(A)$ .

(c)

A sequence $\left\{ x_{n}\right\}$ in $H$ is complete if and only if: $x \in H$ and $\left\langle x, x_{n} \right\rangle = 0$ for every $n$ implies $x = 0$ .

Problem 1.39

(a)(b)

Prove Lemma 1.46.

Origin: Stated according to the original source

(a)

$M \oplus N$ is a closed subspace of $H$ .

(b)

$M \oplus M^{\perp } = H$ .

Problem 1.40

Let $H, K$ be Hilbert spaces. Show that $H \times K$ is a Hilbert space with respect to the inner product $\left\langle \left(h_{1}, k_{1}\right), \left(h_{2}, k_{2}\right) \right\rangle = \left\langle h_{1}, h_{2} \right\rangle_{H}+ \left\langle k_{1}, k_{2} \right\rangle_{K}$ .

Origin: Stated according to the original source

Problem 1.41

Prove that the completion $\widetilde{H}$ (Exercise 1.25) of an inner product space $H$ is a Hilbert space with respect to an inner product that extends the inner product on $H$ .

Origin: Stated according to the original source

Problem 1.42

Prove the remaining parts of Theorems 1.49 and 1.50.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

Theorem 1.49: If $\left\{ x_{n}\right\}$ is an orthonormal sequence in a Hilbert space $H$ , then the following statements hold.

Bessel’s Inequality: $\sum \left| \left\langle x, x_{n} \right\rangle \right|^{2} \leq \left\| x\right\|^{2}$ for every $x \in H$ .
If $x = \sum c_{n} x_{n}$ converges, then $c_{n} = \left\langle x, x_{n} \right\rangle$ .
$\sum c_{n} x_{n}$ converges $\iff$ $\sum \left|c_{n}\right|^{2} < \infty$ .
$x \in \overline{\operatorname {span}}\left\{ x_{n}\right\} \iff x = \sum \left\langle x, x_{n} \right\rangle x_{n}$ .
If $x \in H$ , then $p = \sum \left\langle x, x_{n} \right\rangle x_{n}$ is the orthogonal projection of $x$ onto $\overline{\operatorname {span}}\left\{ x_{n}\right\}$ . Theorem 1.50: If $\left\{ x_{n}\right\}$ is an orthonormal sequence in a Hilbert space $H$ , then the following statements are equivalent.
$\left\{ x_{n}\right\}$ is complete in $H$ .
$\left\{ x_{n}\right\}$ is a basis for $H$ , i.e., for each $x \in H$ there exists a unique sequence of scalars $\left(c_{n}\right)$ such that $x = \sum c_{n} x_{n}$ .
$x = \sum \left\langle x, x_{n} \right\rangle x_{n}$ for each $x \in H$ .
Plancherel’s Equality: $\left\| x\right\|^{2} = \sum \left| \left\langle x, x_{n} \right\rangle \right|^{2}$ for all $x \in H$ .
Parseval’s Equality: $\left\langle x, y \right\rangle = \sum \left\langle x, x_{n} \right\rangle \left\langle x_{n}, y \right\rangle$ for all $x, y \in H$ . The text proves part (a), the implication $\Leftarrow$ of part (c) and the implication $\Rightarrow$ of part (d) of Theorem 1.49, and the implications (a) $\Rightarrow$ (c) and (d) $\Rightarrow$ (c) of Theorem 1.50.

Problem 1.43

(a)(b)

Origin: Stated according to the original source

(a)

Let $M$ be a proper, closed subspace of a Hilbert space $H$ . Given $x \in H \backslash M$ , let $p$ be the orthogonal projection of $x$ onto $M$ , and show that the vector $y = (x-p) /\left\| x-p\right\|$ satisfies $\left\| y\right\| = 1, y \in M^{\perp }$ , and $\operatorname {dist}(y, M) = \inf_{m \in M}\left\| y-m\right\| = 1$ .

(b)

Show that if $H$ is an infinite-dimensional Hilbert space, then there exists an infinite orthonormal sequence $\left\{ e_{n}\right\}$ in $H$ , and no subsequence $\left\{ e_{n_{k}}\right\}$ is Cauchy. Contrast this with the Bolzano-Weierstrass Theorem, which states that every bounded sequence in $\mathbf{F}^{d}$ has a convergent subsequence.

Remark In another language, this problem shows that the closed unit ball $D = \left\{ x \in H:\left\| x\right\| \leq 1\right\}$ in $H$ is not compact when $H$ is infinite dimensional.

Problem 1.44

(a)(b)

This exercise will extend Exercise 1.43 to an arbitrary normed linear space $X$ .

Origin: Stated according to the original source

(a)

Prove F. Riesz's Lemma: If $M$ is a proper, closed subspace of $X$ and $\varepsilon > 0$ , then there exists $x \in X$ with $\left\| x\right\| = 1$ such that $\operatorname {dist}(x, M) = \inf_{m \in M}\left\| x-m\right\| > 1-\varepsilon$ .

(b)

Prove that if $X$ is infinite dimensional then there exists a bounded sequence $\left\{ x_{n}\right\}$ in $X$ that has no convergent subsequences (hence the closed unit ball $D = \left\{ x \in X:\left\| x\right\| \leq 1\right\}$ is never compact in an infinite-dimensional Banach space).

Problem 1.45

Let $\left\{ x_{n}\right\}$ be a finitely linearly independent sequence in a Hilbert space $H$ . Show that there exists an orthogonal sequence $\left\{ y_{n}\right\}$ in $H$ such that $\operatorname {span}\left\{ y_{1}, \ldots , y_{N}\right\} = \operatorname {span}\left\{ x_{1}, \ldots , x_{N}\right\}$ for each $N \in \mathbb {N}$ (this is the GramSchmidt orthogonalization procedure).

Origin: Stated according to the original source

Problem 1.46

(a)(b)

We say that a sequence $\left\{ x_{n}\right\}$ in a Banach space $X$ is $\omega$ -dependent if there exist scalars $c_{n}$ , not all zero, such that $\sum c_{n} x_{n} = 0$ , where the series converges in the norm of $X$ . A sequence is $\omega$ -independent if it is not $\omega$ -dependent (we explore $\omega$ -independent sequences in more detail in Chapter 5).

Origin: Stated according to the original source

(a)

Show that if $\left\{ x_{n}\right\}$ is a basis for a Hilbert space $H$ then $\left\{ x_{n}\right\}$ is $\omega$ independent.

(b)

Let $\alpha , \beta \in \mathbb {C}$ be fixed nonzero scalars such that $\left|\alpha / \beta \right| > 1$ . Let $\left\{ \delta_{n}\right\}_{n \in \mathbb {N}}$ be the standard basis for $\ell^{2}$ , and define $x_{0} = \delta_{1}$ and $x_{n} = \alpha \delta_{n}+\beta \delta_{n+1}$ for $n \in \mathbb {N}$ . Prove that $\left\{ x_{n}\right\}_{n \geq 0}$ is complete and finitely independent in $\ell^{2}$ , but is not $\omega$ -independent and therefore is not a basis for $\ell^{2}$ .

Problem 1.47

(a)(b)

Let $\left\{ x_{n}\right\}$ be a sequence in a Hilbert space $H$ . Prove that the following two statements are equivalent.

Origin: Stated according to the original source

(a)

For each $m \in \mathbb {N}$ we have $x_{m} \notin \overline{\operatorname {span}}\left\{ x_{n}\right\}_{n \neq m}$ (such a sequence is said to be minimal).

(b)

There exists a sequence $\left\{ y_{n}\right\}$ in $H$ such that $\left\langle x_{m}, y_{n} \right\rangle = \delta_{m n}$ for all $m, n \in \mathbb {N}$ (we say that sequences $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ satisfying this condition are biorthogonal).

Show further that, in case these hold, the sequence $\left\{ y_{n}\right\}$ is unique if and only if $\left\{ x_{n}\right\}$ is complete.

Remark Lemma 5.4 will establish an analogous result for Banach spaces.

Problem 1.48

Let $\psi = \chi_{[0,1 / 2)}-\chi_{[1 / 2,1)}$ be the Haar wavelet. Compute the Fourier coefficients of $\psi$ and apply the Plancherel Equality to show that $\frac{\pi^{2}}{8} = \sum_{n = 1}^{\infty } \frac{1}{(2 n-1)^{2}}$ .

Origin: Stated according to the original source

Problem 1.49

This exercise provides a real-valued analogue of the orthonormal basis $\left\{ e^{2 \pi i n t}\right\}_{n \in \mathbb {Z}}$ for complex $L^{2}(\mathbf{T})$ discussed in Example 1.52. Let real $L^{2}(\mathbf{T})$ consist of all real-valued, 1 -periodic functions on $\mathbb {R}$ that are square integrable on $[0,1]$ . Assuming the fact that $\left\{ e^{2 \pi i n t}\right\}_{n \in \mathbb {Z}}$ is complete in complex $L^{2}(\mathbf{T})$ , show that $\left\{ 1\right\} \cup \left\{ \sqrt{2} \sin 2 \pi n t\right\}_{n \in \mathbb {N}} \cup \left\{ \sqrt{2} \cos 2 \pi n t\right\}_{n \in \mathbb {N}}$ forms an orthonormal basis for real $L^{2}(\mathbf{T})$ if we take $\mathbf{F} = \mathbb {R}$ , and forms an orthonormal basis for complex $L^{2}(\mathbf{T})$ if we take $\mathbf{F} = \mathbb {C}$ .

Origin: Stated according to the original source

Problem 1.50

(a)(b)

Origin: Stated according to the original source

(a)

Prove that the Haar system discussed in Remark 1.55(a) is an orthonormal basis for $L^{2}(\mathbb {R})$ .

(b)

Prove that the Haar system given in Remark 1.55(b) is an orthonormal basis for $L^{2}[0,1]$ .

Problem 1.51

(a)(b)

Let $\left\{ x_{n}\right\}$ be an orthonormal basis for a separable Hilbert space $H$ .

Origin: Stated according to the original source

(a)

Show that if $\sum \left\| x_{n}-y_{n}\right\|^{2} < 1$ , then $\left\{ y_{n}\right\}$ is complete in $H$ .

(b)

Show that this conclusion need not hold if $\sum \left\| x_{n}-y_{n}\right\|^{2} = 1$ .

Problem 1.52

Let $\left\{ x_{n}\right\}$ be an orthonormal sequence in a Hilbert space $H$ . Given a vector $x \in H$ , show that $x \in \overline{\operatorname {span}}\left\{ x_{n}\right\}$ if and only if $\left\| x\right\|^{2} = \sum \left| \left\langle x, x_{n} \right\rangle \right|^{2}$ .

Origin: Stated according to the original source

Problem 1.53

Let $M$ be a closed subspace of a Hilbert space $H$ . If $M$ is finite dimensional, let $\operatorname {dim}(M)$ be the dimension of $M$ , otherwise set $\operatorname {dim}(M) = \infty$ . Let $P$ be the orthogonal projection of $H$ onto $M$ . Show that if $\left\{ e_{n}\right\}$ is any orthonormal basis for $H$ , then $\sum \left\| P e_{n}\right\|^{2} = \operatorname {dim}(M)$ .

Origin: Stated according to the original source

Problem 1.54

This result is due to Vitali. Let $\left\{ f_{n}\right\}$ be an orthonormal sequence in $L^{2}[a, b]$ . Show that $\left\{ f_{n}\right\}$ is complete if and only if

\sum _{n = 1}^{\infty }\left|\int _{a}^{x} f_{n}(t) d t\right|^{2} = x-a, \quad x \in [a, b]

Origin: Stated according to the original source

Problem 1.55

(a)(b)

Use the Vitali criterion (Exercise 1.54) to prove that the following two statements are equivalent.

Origin: Stated according to the original source

(a)

The trigonometric system $\left\{ e^{2 \pi i n x}\right\}_{n \in \mathbb {Z}}$ is complete in $L^{2}(\mathbf{T})$ .

(b)

$\sum_{n = 1}^{\infty } \frac{1-\cos 2 \pi n x}{\pi^{2} n^{2}} = x-x^{2}$ for $x \in [0,1]$ .

Remark If we assume Euler's formula

\sum _{n = 1}^{\infty } \frac{1}{n^{2}} = \frac{\pi ^{2}}{6}

then statement (b) reduces to

\sum _{n = 1}^{\infty } \frac{\cos 2 \pi n x}{\pi ^{2} n^{2}} = x^{2}-x+\frac{1}{6}, \quad x \in [0,1] \tag {1.15}

Conversely, equation (1.15) implies Euler's formula by taking $x = 0$ . For a proof that equation (1.15) holds, see Exercise 13.25.

Problem 1.56

This result is due to Dalzell. Let $\left\{ f_{n}\right\}$ be an orthonormal sequence in $L^{2}[a, b]$ . Show that $\left\{ f_{n}\right\}$ is complete if and only if

\sum _{n = 1}^{\infty } \int _{a}^{b}\left|\int _{a}^{x} f_{n}(t) d t\right|^{2} = \frac{(b-a)^{2}}{2}

Origin: Stated according to the original source

Problem 1.57

This result is due to Boas and Pollard [BP48]. Suppose that $\left\{ f_{n}\right\}_{n \in \mathbb {N}}$ is an orthonormal basis for $L^{2}[a, b]$ . Show that there exists a function $m \in L^{\infty }[a, b]$ such that $\left\{ m f_{n}\right\}_{n \geq 2}$ is complete in $L^{2}[a, b]$ .

Origin: Stated according to the original source

Problem 1.58

(a)(b)

Origin: Stated according to the original source

(a)

Let $H$ be a Hilbert space. Show that if $H$ contains an uncountable orthonormal subset, then $H$ is not separable.

(b)

Prove that the space $\ell^{2}(\mathbb {R})$ defined in Example 1.57 is a Hilbert space. Given $t \in \mathbb {R}$ , define $e_{t}(t) = 1$ and $e_{t}(i) = 0$ for $i \neq t$ . Show that $\left\{ e_{t}\right\}_{t \in \mathbb {R}}$ is a complete uncountable orthonormal system for $\ell^{2}(\mathbb {R})$ , and conclude that $\ell^{2}(\mathbb {R})$ is nonseparable.

Problem 1.59

For each $\xi \in \mathbb {R}$ , define a function $e_{\xi }: \mathbb {R} \rightarrow \mathbb {C}$ by $e_{\xi }(t) = e^{2 \pi i \xi t}$ . Let $H = \operatorname {span}\left\{ e_{\xi }\right\}_{\xi \in \mathbb {R}}$ , i.e., $H$ consists of all finite linear combinations of the functions $e_{\xi }$ . Show that

\left\langle f, g \right\rangle = \lim _{T \rightarrow \infty } \frac{1}{2 T} \int _{-T}^{T} f(t) \overline{g(t)} d t, \quad f, g \in H

defines an inner product on $H$ , and $\left\{ e_{\xi }\right\}_{\xi \in \mathbb {R}}$ is an uncountable orthonormal system in $H$ .

Remark The completion $\widetilde{H}$ of $H$ is an important nonseparable Hilbert space. In particular, it contains the class of almost periodic functions [Kat04], and it plays an important role in Wiener's theory of generalized harmonic analysis [Wie33]. Since $\operatorname {span}\left\{ e_{\xi }\right\}_{\xi \in \mathbb {R}} = H$ and $H$ is dense in $\widetilde{H},\left\{ e_{\xi }\right\}_{\xi \in \mathbb {R}}$ is a complete orthonormal system in $\widetilde{H}$ .

Origin: Stated according to the original source

Problem 1.60

Prove Theorem 1.59.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

Theorem 1.59: Let $X, Y$ be normed spaces, and let $f: X \rightarrow Y$ be given. Then $f$ is continuous if and only if $f^{-1}(V)$ is open in $X$ for each open set $V \subseteq Y$ .

Problem 1.61

Prove Theorem 1.61.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

Theorem 1.61: Let $X, Y$ be normed linear spaces, and let $L: X \rightarrow Y$ be a linear operator. (a) $L(0) = 0$ , and $L$ is injective if and only if $\ker L = \left\{ 0\right\}$ ; in particular, if $L$ is an isometry then it is injective. (b) If $L$ is a bijection then the inverse map $L^{-1}: Y \rightarrow X$ is also a linear bijection. (c) $L$ is bounded if and only if $\left\| L\right\| < \infty$ . (d) If $L$ is bounded then $\left\| L x\right\| \leq \left\| L\right\| \left\| x\right\|$ for $x \in X$ , and $\left\| L\right\|$ is the smallest real number $K$ such that $\left\| L x\right\| \leq K\left\| x\right\|$ for all $x \in X$ . (e) $\left\| L\right\| = \sup_{\left\| x\right\| \leq 1}\left\| L x\right\| = \sup_{x \neq 0} \left\| L x\right\| /\left\| x\right\|$ .

Problem 1.62

If $X, Y$ are normed spaces and $L: X \rightarrow Y$ is continuous, show that $\operatorname {ker}(L)$ is a closed subspace of $X$ .

Origin: Stated according to the original source

Problem 1.63

Let $X$ be a normed space and suppose $L \in \mathcal{B}(X)$ . Show that if $\lambda$ is an eigenvalue of $L$ , then $\left|\lambda \right| \leq \left\| L\right\|$ .

Origin: Stated according to the original source

Problem 1.64

(a)(b)(c)

Origin: Stated according to the original source

(a)

Define $L: \ell^{2} \rightarrow \ell^{2}$ by $L(x) = \left(x_{2}, x_{3}, \ldots \right)$ . Prove that this left-shift operator is bounded, linear, surjective, not injective, is not an isometry, and satisfies $\left\| L\right\| = 1$ . Find all of the eigenvalues and eigenvectors of $L$ .

(b)

Define $R: \ell^{2} \rightarrow \ell^{2}$ by $R(x) = \left(0, x_{1}, x_{2}, x_{3}, \ldots \right)$ . Prove that this rightshift operator is bounded, linear, injective, not surjective, and is an isometry. Find all of the eigenvalues and eigenvectors of $R$ .

(c)

Compute $L R$ and $R L$ and show that $L R \neq R L$ . Contrast this with the fact that in finite dimensions, if $A, B: \mathbf{F}^{n} \rightarrow \mathbf{F}^{n}$ are linear maps (hence correspond to multiplication by $n \times n$ matrices), then $A B = I$ if and only if $B A = I$ .

Problem 1.65

(a)(b)

Recall from Exercise 1.22 that $C_{b}(\mathbb {R})$ is a Banach space with respect to the uniform norm $\left\| \cdot \right\|_{\infty }$ , and $C_{b}^{1}(\mathbb {R})$ is a Banach space with respect to the norm $\left\| f\right\|_{C_{b}^{1}} = \left\| f\right\|_{\infty }+\left\| f^{\prime }\right\|_{\infty }$ .

Origin: Stated according to the original source

(a)

Define $D: C_{b}^{1}(\mathbb {R}) \rightarrow C_{b}(\mathbb {R})$ by $D f = f^{\prime }$ , and show that $D$ is a bounded operator.

(b)

Let $D: C_{b}^{1}(\mathbb {R}) \rightarrow C_{b}(\mathbb {R})$ be the same operator as in part (a), but replace the norm on $C_{b}^{1}(\mathbb {R})$ by the $L^{\infty }$ -norm. Show that $D$ is now an unbounded operator.

Problem 1.66

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

Let $\left\{ e_{n}\right\}$ be an orthonormal basis for a separable Hilbert space $H$ , and fix a sequence of scalars $\lambda = \left(\lambda_{n}\right) \in \ell^{\infty }$ . Define

M_{\lambda } x = \sum _{n = 1}^{\infty } \lambda _{n} \left\langle x, e_{n} \right\rangle e_{n}, \quad x \in H,

and prove the following facts.

Origin: Stated according to the original source

(a)

$M_{\lambda }$ is a bounded mapping of $H$ into itself, and its operator norm is $\left\| M_{\lambda }\right\| = \left\| \lambda \right\|_{\ell^{\infty }}$ .

(b)

Each $\lambda_{n}$ is an eigenvalue for $M_{\lambda }$ with corresponding eigenvector $e_{n}$ .

(c)

$M_{\lambda }$ is injective if and only if $\lambda_{n} \neq 0$ for every $n$ .

(d)

$M_{\lambda }$ is surjective if and only if $\inf_{n}\left|\lambda_{n}\right| > 0$ . Further, if $\inf \left|\lambda_{n}\right| = 0$ but $\lambda_{n} \neq 0$ for every $n$ then $\operatorname {range}\left(M_{\lambda }\right)$ is a dense but proper subspace of $H$ .

Problem 1.67

Let $E \subseteq \mathbb {R}$ be Lebesgue measurable, and choose $1 \leq p < \infty$ and $m \in L^{\infty }(\mathbb {R})$ . Define $T_{m}: L^{p}(\mathbb {R}) \rightarrow L^{p}(\mathbb {R})$ by $T_{m} f = f m$ , i.e., pointwise multiplication of $f$ by the function $m$ . Show that $T_{m}$ is bounded and $\left\| T_{m}\right\| = \left\| m\right\|_{\infty }$ .

Origin: Stated according to the original source

Problem 1.68

Prove Theorem 1.66.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

Theorem 1.66: If $V$ is a finite-dimensional vector space and $Y$ is a normed linear space, then any linear operator $T: V \rightarrow Y$ is bounded.

Problem 1.69

(a)(b)(c)

Fill in the details and finish the proof of Theorem 1.67: let $X$ and $Y$ be normed linear spaces.

Origin: Stated according to the original source

(a)

$\mathcal{B}(X,Y)$ is a normed linear space with respect to the operator norm.

(b)

If $Y$ is a Banach space, then $\mathcal{B}(X,Y)$ is a Banach space with respect to the operator norm.

(c)

The operator norm is submultiplicative, i.e., if $A \in \mathcal{B}(X,Y)$ and $B \in \mathcal{B}(Y,Z)$ , then $BA \in \mathcal{B}(X,Z)$ and $\left\| BA\right\| \leq \left\| B\right\| \left\| A\right\|$ .

Problem 1.70

(a)(b)

Let $A$ be an $m \times n$ matrix with entries in $\mathbf{F}$ , which we view as a linear operator $A: \mathbf{F}^{n} \rightarrow \mathbf{F}^{m}$ . The operator norm of $A$ depends on the choice of norm for $\mathbf{F}^{n}$ and $\mathbf{F}^{m}$ .

Origin: Stated according to the original source

(a)

Show that if the norm on $\mathbf{F}^{n}$ and $\mathbf{F}^{m}$ is $\left\| \cdot \right\|_{1}$ then

\left\| A\right\| = \max _{j = 1, \ldots , n}\left\{ \sum _{i = 1}^{m}\left|a_{i j}\right|\right\}

(b)

Show that if the norm on $\mathbf{F}^{n}$ and $\mathbf{F}^{m}$ is $\left\| \cdot \right\|_{\infty }$ then

\left\| A\right\| = \max _{i = 1, \ldots , m}\left\{ \sum _{j = 1}^{n}\left|a_{i j}\right|\right\}

Problem 1.71

Show that if $H, K$ are separable Hilbert spaces, then $H$ and $K$ are isometrically isomorphic.

Origin: Stated according to the original source

Problem 1.72

Let $Y$ be a dense subspace of a normed space $X$ , and let $Z$ be a Banach space. Given $L \in \mathcal{B}(Y, Z)$ , show that there exists a unique operator $\widetilde{L} \in \mathcal{B}(X, Z)$ whose restriction to $Y$ is $L$ . Prove that $\left\| \widetilde{L}\right\| = \left\| L\right\|$ .

Origin: Stated according to the original source

Problem 1.73

Complete the proof of Theorem 1.73 for $p = 1$ and $p = \infty$ .

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

Theorem 1.73 (Dual Space of $\ell^{p}$ ): Fix $1 \leq p \leq \infty$ , and let $p^{\prime }$ be the dual index, $\frac{1}{p}+\frac{1}{p^{\prime }} = 1$ . For each $y \in \ell^{p^{\prime }}$ , let $\mu_{y}: \ell^{p} \rightarrow \mathbf{F}$ be given by $\mu_{y}(x) = \left\langle x, \mu_{y} \right\rangle = \sum_{n} x_{n} y_{n}$ , $x \in \ell^{p}$ . Then the mapping $T: \ell^{p^{\prime }} \rightarrow \left(\ell^{p}\right)^{*}$ given by $T(y) = \mu_{y}$ is a linear isometry of $\ell^{p^{\prime }}$ into $\left(\ell^{p}\right)^{*}$ , and it is an isometric isomorphism if $p < \infty$ .

The text proves the case $1 < p < \infty$ , and has already shown (by Holder's Inequality) that $T$ maps $\ell^{p^{\prime }}$ into $\left(\ell^{p}\right)^{*}$ with $\left\| \mu_{y}\right\| \leq \left\| y\right\|_{\ell^{p^{\prime }}}$ for each $y \in \ell^{p^{\prime }}$ .

Problem 1.74

Let $X$ be a normed space. Given $\mu \in X^{*}, \mu \neq 0$ , show that if $z \notin \operatorname {ker}(\mu )$ , then every $x \in X$ can be written uniquely as $x = y+c z$ where $y \in \operatorname {ker}(\mu )$ and $c \in \mathbf{F}$ .

Origin: Stated according to the original source

Problem 1.75

Given $y = \left(y_{k}\right) \in \ell^{1}$ , define $\mu_{y}: c_{0} \rightarrow \mathbf{F}$ by $\left\langle x, \mu_{y} \right\rangle = \sum x_{k} y_{k}$ . Show that $y \mapsto \mu_{y}$ is an isometric isomorphism of $\ell^{1}$ onto $c_{0}{ }^{*}$ . Thus $c_{0}{ }^{*} \cong \ell^{1}$ .

Origin: Stated according to the original source

Problem 1.76

Let $X$ be a normed space, and suppose that the series $x = \sum x_{n}$ converges in $X$ . Show that if $\mu \in X^{*}$ , then $\sum \left\langle x_{n}, \mu \right\rangle$ is a convergent series of scalars, and $\left\langle x, \mu \right\rangle = \sum \left\langle x_{n}, \mu \right\rangle$ .

Origin: Stated according to the original source

Problem 1.77

(a)(b)(c)

Prove Theorem 1.75: for $h \in H$ , define $\mu_{h}: H \rightarrow \mathbf{F}$ by $\mu_{h}(f) = \left\langle f, h \right\rangle$ .

Origin: Stated according to the original source

(a)

$\mu_{h} \in H^{*}$ and $\left\| \mu_{h}\right\| = \left\| h\right\|$ for each $h \in H$ .

(b)

For every $\mu \in H^{*}$ , there exists $h \in H$ such that $\mu = \mu_{h}$ .

(c)

Conclude that the mapping $h \mapsto \mu_{h}$ is an antilinear bijective isometry of $H$ onto $H^{*}$ .

Problem 1.78

Fix $1 \leq p \leq \infty$ , and let $y = \left(y_{n}\right)$ be a fixed sequence of scalars. Use Theorem 1.73 to show that there exists a constant $C$ such that $\left| \left\langle x, y \right\rangle \right| \leq C\left\| x\right\|_{\ell^{p}}$ for all finite sequences $x \in c_{00}$ if and only if $y \in \ell^{p^{\prime }}$ .

Origin: Stated according to the original source

Problem 1.79

Let $\mu$ be a linear functional on a normed space $X$ . Prove that

\mu \text{ is bounded } \Longleftrightarrow \operatorname {ker}(\mu ) \text{ is closed. }

Origin: Stated according to the original source

Problem 1.80

Let $X, Y$ be normed spaces. Then, by Exercise 1.16, the Cartesian product $X \times Y$ is a normed space with respect to the norm $\left\| (x, y)\right\|_{\infty } = \max \left\{ \left\| x\right\| ,\left\| y\right\| \right\}$ . Likewise, $X^{*} \times Y^{*}$ is normed with respect to $\left\| \left(x^{*}, y^{*}\right)\right\|_{1} = \left\| x^{*}\right\| +\left\| y^{*}\right\|$ . Show that, with respect to these choices of norms, $(X \times Y)^{*}$ is isometrically isomorphic to $X^{*} \times Y^{*}$ .

Origin: Stated according to the original source

Chapter 2Functional Analysis[52/100%]

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 $\left\| \Lambda \right\| = \left\| \lambda \right\|$ .

Origin: Stated according to the original source

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 $\left\| \mu \right\| = 1 / d$ .

Origin: Stated according to the original source

Problem 2.3

(a)(b)

Let $X$ be a normed space.

Origin: Stated according to the original source

(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 $C[0,1]$ .

Origin: Stated according to the original source

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.

Origin: Stated according to the original source

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\} .

Origin: Stated according to the original source

Problem 2.7

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

Origin: Stated according to the original source

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 $\left(\ell^{\infty }\right)^{*}$ .

Origin: Stated according to the original source

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).

Origin: Stated according to the original source

Problem 2.10

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

Origin: Stated according to the original source

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.

Origin: Stated according to the original source

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 $\epsilon^{*} \mu = \left.\mu \right|_{M}$ .

Origin: Stated according to the original source

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 $A^{\mathrm{T}}$ .

Origin: Stated according to the original source

Problem 2.14

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

Origin: Stated according to the original source

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.

Origin: Stated according to the original source

Problem 2.16

Prove Theorem 2.13.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

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 $(P) = M$ .

Origin: Stated according to the original source

Problem 2.18

(a)(b)(c)

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

Origin: Stated according to the original source

(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.

Origin: Stated according to the original source

(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.

Origin: Stated according to the original source

Problem 2.21

(a)(b)

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

Origin: Stated according to the original source

(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 $U^{-1} = U^{*}$ .

Origin: Stated according to the original source

Problem 2.23

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

Fill in the details in the proof of Theorem 2.18.

Origin: Stated according to the original source

(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 $H$ .

Origin: Stated according to the original source

(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.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

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 $C_{0}(\mathbb {R})$ .

Origin: Stated according to the original source

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 $t \in (a, b)$ .

Origin: Stated according to the original source

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.

Origin: Stated according to the original source

Problem 2.29

Prove Theorem 2.23.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

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 $N \rightarrow \infty$ .

Origin: Stated according to the original source

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 $X$ .

Origin: Stated according to the original source

(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 $p = \infty$ .

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

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)

Origin: Stated according to the original source

(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

Origin: Stated according to the original source

(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 $Y$ .

Origin: Stated according to the original source

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 $Y$ .

Origin: Stated according to the original source

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 $x \in X$ .

Origin: Stated according to the original source

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.

Origin: Stated according to the original source

(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.

Origin: Stated according to the original source

(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 $(I-T)^{-1}$ ).

Origin: Stated according to the original source

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.

Origin: Stated according to the original source

Problem 2.42

(a)(b)

Prove Theorem 2.31.

Origin: Stated according to the original source

(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.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

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.

Origin: Stated according to the original source

(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.

Origin: Stated according to the original source

(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.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

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.

Origin: Stated according to the original source

Problem 2.48

(a)(b)

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

Origin: Stated according to the original source

(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 $f^{\prime } = g$ .

Origin: Stated according to the original source

Problem 2.50

Prove part (b) of Theorem 2.38.

Origin: Stated according to the original source

Problem 2.51

(a)(b)

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

Origin: Stated according to the original source

(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 $x_{n} \xrightarrow {\mathrm{w}} 0$ .

Origin: Stated according to the original source