Hilbert Spaces

Let $V$ denote the vector space of bounded continuous functions from $\mathbb {R}$ to $\mathbf{F}$. Let $r_{1}, r_{2}, \ldots$ be a list of the rational numbers. For $f, g \in V$, define

Show that $\left\langle \cdot , \cdot \right\rangle$ is an inner product on $V$.

Prove that if $\mu$ is a measure and $f, g \in L^{2}(\mu )$, then

Suppose $f$ and $g$ are elements of an inner product space and

Find $a, b \in \mathbb {R}^{3}$ such that $a$ is a scalar multiple of $(1,6,3), b$ is orthogonal to $(1,6,3)$, and $(5,4,-2) = a+b$.

Prove that

for all positive numbers $a, b, c, d$, with equality if and only if $a = b = c = d$.

Prove that the square of the average of each finite list of real numbers containing at least two distinct real numbers is less than the average of the squares of the numbers in that list.

Suppose $f$ and $g$ are elements of an inner product space and $\left\| f\right\| \leq 1$ and $\left\| g\right\| \leq 1$. Prove that

Suppose $a$ and $b$ are nonzero elements of $\mathbb {R}^{2}$. Prove that

where $\theta$ is the angle between $a$ and $b$ (thinking of $a$ as the vector whose initial point is the origin and whose end point is $a$, and similarly for $b$).

The angle between two vectors (thought of as arrows with initial point at the origin) in $\mathbb {R}^{2}$ or $\mathbb {R}^{3}$ can be defined geometrically. However, geometry is not as clear in $\mathbb {R}^{n}$ for $n > 3$. Thus the angle between two nonzero vectors $a, b \in \mathbb {R}^{n}$ is defined to be

where the motivation for this definition comes from the previous exercise. Explain why the Cauchy-Schwarz inequality is needed to show that this definition makes sense.

Suppose $f$ and $g$ are elements of an inner product space. Prove that $\left\| f\right\| = \left\| g\right\|$ if and only if $\left\| sf + tg\right\| = \left\| t f+s g\right\|$ for all $s, t \in \mathbb {R}$.

Suppose $f$ and $g$ are elements of an inner product space and $\left\| f\right\| = \left\| g\right\| = 1$ and $\left\langle f, g \right\rangle = 1$. Prove that $f = g$.

Suppose $f$ and $g$ are elements of a real inner product space. Prove that

Suppose $f$ and $g$ are elements of a complex inner product space. Prove that

Suppose $f, g, h$ are elements of an inner product space. Prove that

Prove that a norm satisfying the parallelogram equality comes from an inner product. In other words, show that if $V$ is a normed vector space whose norm $\left\| \cdot \right\|$ satisfies the parallelogram equality, then there is an inner product $\left\langle \cdot , \cdot \right\rangle$ on $V$ such that $\left\| f\right\| = \left\langle f, f \right\rangle^{1 / 2}$ for all $f \in V$.

Let $\lambda$ denote Lebesgue measure on $[1, \infty )$.

Suppose $\mu$ is a measure. For $f, g \in L^{2}(\mu )$, define $\left\langle f, g \right\rangle$ by

Suppose $V_{1}, \ldots , V_{m}$ are inner product spaces. Show that the equation

defines an inner product on $V_{1} \times \cdots \times V_{m}$.

Suppose $V$ is an inner product space. Make $V \times V$ an inner product space as in the exercise above. Prove that the function that takes an ordered pair $(f, g) \in V \times V$ to the inner product $\left\langle f, g \right\rangle \in \mathbf{F}$ is a continuous function from $V \times V$ to $\mathbf{F}$.

Suppose $1 \leq p \leq \infty$.

Use inner products to prove Apollonius's identity: In a triangle with sides of length $a, b$, and $c$, let $d$ be the length of the line segment from the midpoint of the side of length $c$ to the opposite vertex. Then

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

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

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

Show that the equation

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

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

and

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

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

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.

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

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

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

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

for all $h \in U$.

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

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.

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 $\left\| a-b\right\| = \operatorname {distance}(a, U)$.

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 $\left\| z-w\right\|_{\infty } = \operatorname {distance}(z, U)$.

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

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

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 $f \in U$.

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 $P = P_{U}$.

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 $\left\| T\right\| = \left\| S\right\|$.

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

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 $g \in W$.

Verify the assertions in Example 8.46.

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

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

for all $f \in V$.

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

Verify that the family $\left\{ e_{k}\right\}_{k \in \mathbf{Z}}$ as defined in the third bullet point of Example 8.51 is an orthonormal family in $L^{2}((-\pi , \pi ])$. The following formulas should help:

Suppose $\left\{ a_{k}\right\}_{k \in \Gamma }$ is a family in $\mathbb {R}$ and $a_{k} \geq 0$ for each $k \in \Gamma$. Prove the unordered sum $\sum_{k \in \Gamma } a_{k}$ converges if and only if

Furthermore, prove that if $\sum_{k \in \Gamma } a_{k}$ converges then it equals the supremum above.

Suppose $\left\{ e_{k}\right\}_{k \in \Gamma }$ is an orthonormal family in an inner product space $V$. Prove that if $f \in V$, then $\left\{ k \in \Gamma : \left\langle f, e_{k} \right\rangle \neq 0\right\}$ is a countable set.

Suppose $\left\{ f_{k}\right\}_{k \in \Gamma }$ and $\left\{ g_{k}\right\}_{k \in \Gamma }$ are families in a normed vector space such that $\sum_{k \in \Gamma } f_{k}$ and $\sum_{k \in \Gamma } g_{k}$ converge. Prove that $\sum_{k \in \Gamma }\left(f_{k}+g_{k}\right)$ converges and

Suppose $\left\{ f_{k}\right\}_{k \in \Gamma }$ is a family in a normed vector space such that $\sum_{k \in \Gamma } f_{k}$ converges. Prove that if $c \in \mathbf{F}$, then $\sum_{k \in \Gamma }\left(c f_{k}\right)$ converges and

Suppose $\left\{ a_{k}\right\}_{k \in \Gamma }$ is a family in $\mathbb {R}$. Prove that the unordered sum $\sum_{k \in \Gamma } a_{k}$ converges if and only if $\sum_{k \in \Gamma }\left|a_{k}\right| < \infty$.

Suppose $\left\{ f_{k}\right\}_{k \in \mathbb {N}}$is a family in a normed vector space $V$ and $f \in V$. Prove that the unordered sum $\sum_{k \in \mathbb {N}} f_{k}$ equals $f$ if and only if the usual ordered sum $\sum_{k = 1}^{\infty } f_{p(k)}$ equals $f$ for every injective and surjective function $p: \mathbb {N} \rightarrow \mathbb {N}$.

Explain why 8.58 implies that if $\Gamma$ is a finite set and $\left\{ e_{k}\right\}_{k \in \Gamma }$ is an orthonormal family in a Hilbert space $V$, then $\operatorname {span}\left\{ e_{k}\right\}_{k \in \Gamma }$ is a closed subspace of $V$.

Suppose $V$ is an infinite-dimensional Hilbert space. Prove that there does not exist a basis of $V$ that is an orthonormal family.

Suppose $\mu$ is a $\sigma$-finite measure on $(X, \mathcal{S})$ and $\nu$ is a $\sigma$-finite measure on $(Y, \mathcal{T})$. Suppose also that $\left\{ e_{j}\right\}_{j \in \Omega }$ is an orthonormal basis of $L^{2}(\mu )$ and $\left\{ f_{k}\right\}_{k \in \Gamma }$ is an orthonormal basis of $L^{2}(\nu )$ for some countable set $\Gamma$. For $j \in \Omega$ and $k \in \Gamma$, define $g_{j, k}: X \times Y \rightarrow \mathbf{F}$ by

Prove that $\left\{ g_{j, k}\right\}_{j \in \Omega , k \in \Gamma }$ is an orthonormal basis of $L^{2}(\mu \times \nu )$.

Prove the converse of Parseval's identity. More specifically, prove that if $\left\{ e_{k}\right\}_{k \in \Gamma }$ is an orthonormal family in a Hilbert space $V$ and

for every $f \in V$, then $\left\{ e_{k}\right\}_{k \in \Gamma }$ is an orthonormal basis of $V$.

Prove that every subspace of a separable normed vector space is separable.

Suppose $V$ is an infinite-dimensional Hilbert space. Prove that there does not exist a translation invariant measure on the Borel subsets of $V$ that assigns positive but finite measure to each open ball in $V$.

Find the polynomial $g$ of degree at most $4$ that minimizes $\int_{0}^{1}\left|x^{5}-g(x)\right|^{2} d x$.

Prove that each orthonormal family in a Hilbert space can be extended to an orthonormal basis of the Hilbert space. Specifically, suppose $\left\{ e_{j}\right\}_{j \in \Omega }$ is an orthonormal family in a Hilbert space $V$. Prove that there exists a set $\Gamma$ containing $\Omega$ and an orthonormal basis $\left\{ f_{k}\right\}_{k \in \Gamma }$ of $V$ such that $f_{j} = e_{j}$ for every $j \in \Omega$.

Prove that every vector space has a basis.

Find the polynomial $g$ of degree at most 4 such that

for every polynomial $f$ of degree at most 4.

Suppose $G$ is a nonempty open subset of C. The Bergman space $L_{a}^{2}(G)$ is defined to be the set of analytic functions $f: G \rightarrow \mathbb {C}$ such that

where $\lambda_{2}$ is the usual Lebesgue measure on $\mathbb {R}^{2}$, which is identified with $\mathbb {C}$. For $f, h \in L_{a}^{2}(G)$, define $\left\langle f, h \right\rangle$ to be $\int_{G} f \overline{h} d \lambda_{2}$.

Let $\mathbf{D}$ denote the open unit disk in $\mathbb {C}$; thus

Suppose $G$ is the annulus defined by

Prove that if $f \in L_{a}^{2}(\mathbf{D} \backslash \left\{ 0\right\} )$, then $f$ has a removable singularity at 0 (meaning that $f$ can be extended to a function that is analytic on $\mathbf{D}$).

The Dirichlet space $\mathcal{D}$ is defined to be the set of analytic functions $f: \mathbf{D} \rightarrow \mathbb {C}$ such that