Math Problems Vault

Problem 8B.1

Show that each of the inner product spaces in Example 8.23 is not a Hilbert space.

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Remark.

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Example 8.23:

The inner product space $\ell^{1}$ , where $\left\langle (a_1,a_2,\ldots ), (b_1,b_2,\ldots ) \right\rangle = \sum_{k=1}^{\infty } a_k b_k$ , is not a Hilbert space because the associated norm is not complete on $\ell^{1}$ .
The inner product space $C([0,1])$ of continuous $\mathbf{F}$ -valued functions on the interval $[0,1]$ , where $\left\langle f, g \right\rangle = \int_0^1 fg$ , is not a Hilbert space because the associated norm is not complete on $C([0,1])$ .

Problem 8B.2

Prove or disprove: The inner product space in Exercise 1 in Section 8A is a Hilbert space.

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Problem 8B.3

Suppose $V_{1}, V_{2}, \ldots$ are Hilbert spaces. Let

V = \left\{ \left(f_{1}, f_{2}, \ldots \right) \in V_{1} \times V_{2} \times \cdots : \sum _{k = 1}^{\infty }\left\| f_{k}\right\| ^{2} < \infty \right\} .

Show that the equation

\left\langle \left(f_{1}, f_{2}, \ldots \right), \left(g_{1}, g_{2}, \ldots \right) \right\rangle = \sum _{k = 1}^{\infty } \left\langle f_{k}, g_{k} \right\rangle

defines an inner product on $V$ that makes $V$ a Hilbert space.

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Remark.

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Each of the Hilbert spaces $V_{1}, V_{2}, \ldots$ may have a different inner product, even though the same notation is used for the norm and inner product on all these Hilbert spaces.

Problem 8B.4

Suppose $V$ is a real Hilbert space. The complexification of $V$ is the complex vector space $V_{\mathbb {C}}$ defined by $V_{\mathbb {C}} = V \times V$ , but we write a typical element of $V_{\mathbb {C}}$ as $f+i g$ instead of $(f, g)$ . Addition and scalar multiplication are defined on $V_{\mathbb {C}}$ by

\left(f_{1}+i g_{1}\right)+\left(f_{2}+i g_{2}\right) = \left(f_{1}+f_{2}\right)+i\left(g_{1}+g_{2}\right)

and

(\alpha +i \beta )(f+i g) = (\alpha f-\beta g)+i(\alpha g+\beta f)

for $f_{1}, f_{2}, f, g_{1}, g_{2}, g \in V$ and $\alpha , \beta \in \mathbb {R}$ . Show that

\left\langle f_{1}+i g_{1}, f_{2}+i g_{2} \right\rangle = \left\langle f_{1}, f_{2} \right\rangle + \left\langle g_{1}, g_{2} \right\rangle +i\left( \left\langle g_{1}, f_{2} \right\rangle - \left\langle f_{1}, g_{2} \right\rangle \right)

defines an inner product on $V_{\mathbb {C}}$ that makes $V_{\mathbb {C}}$ into a complex Hilbert space.

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Problem 8B.5

Prove that if $V$ is a normed vector space, $f \in V$ , and $r > 0$ , then the open ball $B(f, r)$ centered at $f$ with radius $r$ is convex.

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Problem 8B.6

(a)(b)

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(a)

Suppose $V$ is an inner product space and $B$ is the open unit ball in $V$ (thus $B = \left\{ f \in V:\left\| f\right\| < 1\right\}$ ). Prove that if $U$ is a subset of $V$ such that $B \subseteq U \subseteq \overline{B}$ , then $U$ is convex.

(b)

Give an example to show that the result in part (a) can fail if the phrase inner product space is replaced by Banach space.

Problem 8B.7

Suppose $V$ is a normed vector space and $U$ is a closed subset of $V$ . Prove that $U$ is convex if and only if

\frac{f+g}{2} \in U \text{ for all } f, g \in U

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Problem 8B.8

Prove that if $U$ is a convex subset of a normed vector space, then $\overline{U}$ is also convex.

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Problem 8B.9

Prove that if $U$ is a convex subset of a normed vector space, then the interior of $U$ is also convex.

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Remark.

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The interior of $U$ is the set $\left\{ f \in U: B(f, r) \subseteq U\text{ for some }r > 0\right\}$ .

Problem 8B.10

Suppose $V$ is a Hilbert space, $U$ is a nonempty closed convex subset of $V$ , and $g \in U$ is the unique element of $U$ with smallest norm (obtained by taking $f = 0$ in 8.28). Prove that

\operatorname {Re} \left\langle g, h \right\rangle \geq \left\| g\right\| ^{2}

for all $h \in U$ .

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Problem 8B.11

Suppose $V$ is a Hilbert space. A closed half-space of $V$ is a set of the form

\left\{ g \in V: \operatorname {Re} \left\langle g, h \right\rangle \geq c\right\}

for some $h \in V$ and some $c \in \mathbb {R}$ . Prove that every closed convex subset of $V$ is the intersection of all the closed half-spaces that contain it.

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Problem 8B.12

Give an example of a nonempty closed subset $U$ of the Hilbert space $\ell^{2}$ and $a \in \ell^{2}$ such that there does not exist $b \in U$ with $\left\| a-b\right\| = \operatorname {distance}(a, U)$ .

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Remark.

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By 8.28, U cannot be a convex subset of $\ell^{2}$ .

Problem 8B.13

In the real Banach space $\mathbb {R}^{2}$ with norm defined by $\left\| (x, y)\right\|_{\infty } = \max \left\{ \left|x\right|,\left|y\right|\right\}$ , give an example of a closed convex set $U \subseteq \mathbb {R}^{2}$ and $z \in \mathbb {R}^{2}$ such that there exist infinitely many choices of $w \in U$ with $\left\| z-w\right\|_{\infty } = \operatorname {distance}(z, U)$ .

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Problem 8B.14

Suppose $f$ and $g$ are elements of an inner product space. Prove that $\left\langle f, g \right\rangle = 0$ if and only if

\left\| f\right\| \leq \left\| f+\alpha g\right\|

for all $\alpha \in \mathbf{F}$ .

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Problem 8B.15

Suppose $U$ is a closed subspace of a Hilbert space $V$ and $f \in V$ . Prove that $\left\| P_{U} f\right\| \leq \left\| f\right\|$ , with equality if and only if $f \in U$ .

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Remark.

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This exercise asks you to prove $8.37(d)$ .

Problem 8B.16

Suppose $V$ is a Hilbert space and $P: V \rightarrow V$ is a linear map such that $P^{2} = P$ and $\left\| P f\right\| \leq \left\| f\right\|$ for every $f \in V$ . Prove that there exists a closed subspace $U$ of $V$ such that $P = P_{U}$ .

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Problem 8B.17

Suppose $U$ is a subspace of a Hilbert space $V$ . Suppose also that $W$ is a Banach space and $S: U \rightarrow W$ is a bounded linear map. Prove that there exists a bounded linear map $T: V \rightarrow W$ such that $\left.T\right|_{U} = S$ and $\left\| T\right\| = \left\| S\right\|$ .

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Remark.

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If $W = \mathbf{F}$ , then this result is just the Hahn-Banach Theorem (6.69) for Hilbert spaces. The result here is stronger because it allows $W$ to be an arbitrary Banach space instead of requiring $W$ to be $\mathbf{F}$ . Also, the proof in this Hilbert space context does not require use of Zorn's Lemma or the Axiom of Choice.

Problem 8B.18

Suppose $U$ and $W$ are subspaces of a Hilbert space $V$ . Prove that $\overline{U} = \overline{W}$ if and only if $U^{\perp } = W^{\perp }$ .

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Problem 8B.19

Suppose $U$ and $W$ are closed subspaces of a Hilbert space. Prove that $P_{U} P_{W} = 0$ if and only if $\left\langle f, g \right\rangle = 0$ for all $f \in U$ and all $g \in W$ .

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Problem 8B.20

Verify the assertions in Example 8.46.

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Remark.

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Example 8.46: suppose $U$ is the closed subspace of $L^{2}(\mathbb {R})$ defined by $U = \left\{ f \in L^{2}(\mathbb {R}) : f(x) = 0 \text{ for almost every } x < 0\right\}$ . Then $U^{\perp } = \left\{ g \in L^{2}(\mathbb {R}) : g(x) = 0 \text{ for almost every } x \geq 0\right\}$ . Furthermore, if $h \in L^{2}(\mathbb {R})$ , then $P_{U} h = h \chi_{[0,\infty )}$ and $P_{U^{\perp }} h = h \chi_{(-\infty ,0)}$ .

Problem 8B.21

Show that every inner product space is a subspace of some Hilbert space.

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Problem 8B.22

Prove that if $V$ is a Hilbert space and $T: V \rightarrow V$ is a bounded linear map such that the dimension of range $T$ is $1$ , then there exist $g, h \in V$ such that

T f = \left\langle f, g \right\rangle h

for all $f \in V$ .

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Problem 8B.23

(a)(b)

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(a)

Give an example of a Banach space $V$ and a bounded linear functional $\varphi$ on $V$ such that $\left|\varphi (f)\right| < \left\| \varphi \right\| \left\| f\right\|$ for all $f \in V \backslash \left\{ 0\right\}$ .

(b)

Show there does not exist an example in part (a) where $V$ is a Hilbert space.

Problem 8B.24

(a)(b)

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(a)

(b)

Give an example to show that part (a) can fail if the hypothesis that $V$ is a Hilbert space is replaced by the hypothesis that $V$ is a Banach space.

Problem 8B.25

(a)(b)

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(a)

Suppose that $\mu$ is a finite measure, $1 \leq p \leq 2$ , and $\varphi$ is a bounded linear functional on $L^{p}(\mu )$ . Prove that there exists $h \in L^{p^{\prime }}(\mu )$ such that $\varphi (f) = \int f h d \mu$ for every $f \in L^{p}(\mu )$ .

(b)

Same as part (a), but with the hypothesis that $\mu$ is a finite measure replaced by the hypothesis that $\mu$ is a measure, and assume that $1 < p \leq 2$ .

Remark.

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See 7.25, which along with this exercise shows that we can identify the dual of $L^{p}(\mu )$ with $L^{p^{\prime }}(\mu )$ for $1 < p \leq 2$ . See 9.42 for an extension to all $p \in (1, \infty )$ .

Problem 8B.26

Prove that if $V$ is an infinite-dimensional Hilbert space, then the Banach space $\mathcal{B}(V, V)$ is nonseparable.

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