**Exercise 7.1** Let *H* be a Hilbert space and suppose *x* and *y* are nonzero elements of *H*. Show that \(\|x+y\| = \|x\|+\|y\|\) if and only if *y = cx*, where \(c>0\).

**Exercise 7.2** Let *H* be a Hilbert space. Suppose *T* and *S* are operators on *H* and let \(\alpha\in \mathbb{C}\). Show that \((T+S)^\ast = T^\ast + S^\ast\), \((\alpha T)^\ast = \overline{\alpha}\, T^\ast\), and \((ST)^\ast = T^\ast S^\ast\).

**Exercise 7.3** Suppose *T* and *S* are hermitian operators on a Hilbert space. Show that *TS* is hermitian if and only if *TS = ST*.

**Exercise 7.4** Let *H* be a normed space that satisfies the Parallelogram Law (Theorem 7.10). Show that *H* is an inner product space. (*Hint:* Use the polarization formulas given after the statement of Theorem 7.10.)

**Exercise 7.5** Let *H* be a complex inner product space and assume \(A:H\rightarrow H\) is a bounded linear operator. Show that \((Ax,y)\) can be written as

$$\frac{1}{4}\bigg[ \Big(A(x+y), x+y\Big) - \Big(A(x-y), x-y\Big) - i\Big(A(x+iy), x+iy\Big) + i\Big(A(x-iy), x-iy\Big) \bigg].$$

**Exercise 7.6** Let

*H* be an inner product space and assume

*A* and

*B* are bounded linear operators on

*H*.

- (a)
Show that if \((Ax, y) = (Bx, y)\) for all *x* and *y* in *H*, then *A = B*.

- (b)
Show that if \((Ax,x) = (Bx,x)\) for all \(x\in H\) and *H* is a *complex* inner product space, then *A = B*.

- (c)
What assumptions need to be added to *A* and *B* in order for (b) to hold when *H* is a *real* inner product space?

**Exercise 7.7** Suppose *A* is a bounded linear operator on the complex inner product space *H*. Show that

$$\|A\|=\sup_{x\in B_H}|(Ax, x)|.$$

Show that the same formula holds in a real inner product space if *A* is hermitian.

**Exercise 7.8** Let *H* be a complex Hilbert space and let \(T:H\rightarrow H\) be a bounded linear operator. Show that \(T=T^\ast\) if and only if \((Tx,x)\in\mathbb{R}\) for all \(x\in H\). (This equivalence cannot hold in a real Hilbert space because it is necessarily true that \((Tx,y)\in\mathbb{R}\) for all *x* and *y* in *H* when *H* is a real Hilbert space.)

**Exercise 7.9** Let *H* be an inner product space and assume \(T:H\rightarrow H\) is a bounded linear operator. Show that \(\|T^\ast T\| = \|T\|^2\).

**Exercise 7.10** Let \(T: H\rightarrow H\) be such that \(T(0)=0\) and \(\|T(x) - T(y)\|= \|x-y\|\) for all *x* and *y* in *H*. Show that *T* is a linear isometry from *H* to itself. (*Hint:* Show first that \(\big(T(x), T(y)\big)=(x, y)\) for all *x* and *y* in *H*).

**Exercise 7.11** Let *H* be a Hilbert space. For each \(\phi\in H^\ast\), let \(v_\phi \in H\) be the unique element (from the Riesz–Fr échet Theorem) satisfying \(\phi(x)=(x, v_\phi)\) for all \(x\in H\). If \(T_O^\ast:H^\ast\rightarrow H^\ast\) is the operator adjoint of *T* (see Definition 3.36) and \(T_A^\ast:H\rightarrow H\) is the Hilbert space adjoint of *T* (see Definition 7.18), show that \(v_{T_O^\ast(\phi)} = T_A^\ast v_\phi\) for all \(\phi\in H^\ast\).

**Exercise 7.12** If *V* is a closed subspace of a Hilbert space *H*, show \(H = V \oplus V^\perp\).

**Exercise 7.13** If *V* is a closed subspace of a Hilbert space *H*, show \((V^\perp)^\perp = V\). What is \((V^\perp)^\perp\) if *V* is not closed?

**Exercise 7.14** Let *H* be a Hilbert space and suppose \((x_n)_{n=1}^\infty\) and \((y_n)_{n=1}^\infty\) are sequences in *H*. If \((x_n)_{n=1}^\infty\) and \((y_n)_{n=1}^\infty\) converge (in norm) to *x* and *y*, respectively, show that \(\lim_{n\rightarrow\infty}(x_n, y_n) = (x,y)\).

**Exercise 7.15** **(Hellinger–Toeplitz Theorem)** Let *H* be a Hilbert space. Prove the following: If \(T:H\rightarrow H\) is a linear map that satisfies the equation \((Tx,y)=(x,Ty)\) for all *x* and *y* in *H*, then *T* is continuous.

**Exercise 7.16** Let *H* be a Hilbert space with inner product \((\cdot, \cdot)\) and norm \(\|\cdot\|\). If \((x_n)_{n=1}^\infty\) and \((y_n)_{n=1}^\infty\) are sequences in *B* _{ H }, and \(\lim_{n\rightarrow\infty} (x_n,y_n) =1\), show that \(\lim_{n\rightarrow\infty} \|x_n-y_n\|=0\).

**Exercise 7.17** Let *H* be an infinite-dimensional Hilbert space with inner product \((\cdot,\cdot)\). Show that the function \((\cdot,\cdot):(H,w)\times(H,w)\rightarrow\mathbb{C}\) is continuous in each argument separately, but that it is not continuous on the product \((H,w)\times(H,w)\). (In this problem, \((H,w)\) denotes the Hilbert space *H* endowed with the weak topology.)

**Exercise 7.18** Solve the system of differential equations

$$\begin{cases} y'' + \lambda^2 y = 0, & \lambda\in\mathbb{R}, \\ y(0) = 1, y(2\pi) = 1. &\end{cases}$$

Use your answer to show that \((e_n)_{n\in\mathbb{Z}}\) is an orthonormal basis for \(L_2\Big(\mathbb{T}, \frac{d\theta}{2\pi}\Big)\), where \(e_n(\theta)=e^{in\theta}\) and \(\theta\in\mathbb{T}=[0,2\pi)\).

**Exercise 7.19** Use Parseval’s Identity (Theorem 7.25) to show that

\(\displaystyle\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}\):

- (a)
Use the function \(f(x)=x\) for all \(x\in [0,\pi)\) and Theorem 7.43.

- (b)
Use the function \(f(\theta)=\theta\) for all \(\theta\in [0, 2\pi)\) and Exercise 7.18.

**Exercise 7.20** A theorem from linear algebra states that the trace of a square matrix equals the sum of its eigenvalues. If

\(T_K: L_2(a,b)\rightarrow L_2(a,b)\) is a Hilbert–Schmidt operator defined by the formula

\(T_K f(x)=\int_a^b K(x,y) f(y) \, dy\), then the

*trace of the Hilbert–Schmidt operator T* _{ K } is defined to be

$$\mbox{trace}(T_K) = \int_a^b K(x,x) \, dx,$$

whenever it exists. Let *K* be the kernel in (7.23) and show \(\mbox{trace}(T_K) = \sum_{n=1}^\infty \lambda_n\), where \((\lambda_n)_{n=1}^\infty\) is the sequence of eigenvalues for *T* _{ K } given in (7.25). (Compare to (6.1.7).)

**Exercise 7.21** Compute the integral in (7.26) to show that \(\displaystyle\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}\).

**Exercise 7.22** Suppose μ and ν are probability measures on a measure space \((\Omega, \Sigma)\). Assume that \(\mu \ll \nu\) and *ϕ* is the Radon–Nikodým derivative of μ with respect to ν. Define a map \(V: L_2(\mu)\rightarrow L_2(\nu)\) by \(V(f) = \sqrt{\phi} f\) for all \(f\in L_2(\mu)\). Show that this map is a well-defined isometry. Show that *V* is an isomorphism if and only if \(\nu \ll \mu\).

**Exercise 7.23** Recall that a function

\(f:[0,1]\rightarrow\mathbb{R}\) is called

*absolutely continuous* on [0,1] if

*f* is differentiable almost everywhere (with respect to Lebesgue measure), and if

\(f'\in L_1(0,1)\) satisfies the equation

\(f(x)-f(0) = \int_0^x f'(t)\, {\it dt}\) for all

\(x\in [0,1]\).

- (a)
Let *H* denote the collection of all (real-valued) absolutely continuous functions on [0,1] such that \(f(0)=0\) and \(f'\in L_2(0,1)\). Show that

$$(f,g) = \int_0^1 f'(t)\, g'(t)\, {\it dt}, \{f, g\} \subseteq H,$$

defines a complete inner product on *H*.

- (b)
Show that the map \(T:H\rightarrow L_2(0,1)\), defined by \(Tf = f'\) for all \(f\in H\), is an isomorphism. Find *T* ^{-1}.

- (c)
Fix \(a\in (0,1)\) and define a map \(\Lambda_a:H\rightarrow \mathbb{R}\) by \(\Lambda_a(f) = f(a)\) for all \(f\in H\). Show that Λ_{ a } is a bounded linear functional. Find the element \(\phi_a\in H\) such that \(\Lambda_a(f) = (f,\phi_a)\) for all \(f\in H\).

**Exercise 7.24** Let *H* be a Hilbert space and suppose \(T: H \rightarrow H\) is a compact operator. Show that *T* is the limit (in operator norm) of a sequence of finite-rank operators.

**Exercise 7.25** A Banach space *X* is called *uniformly convex* if given \(\epsilon>0\) there exists a \(\delta>0\) such that \(\|x-y\|<\epsilon\) whenever \(\|x\|\leq 1\), \(\|y\|\leq 1\), and \(\|x+y\|>2-\delta\). Show that a Hilbert space is uniformly convex.

**Exercise 7.26** Suppose

*X* is a non-reflexive Banach space and let

\(\epsilon>0\) be given.

- (a)
Show there exists an \(x^{\ast\ast}\in X^{\ast\ast}\) such that \(\|x^{\ast\ast}\|=1\) and

$$d(x^{\ast\ast}, X) = \inf\{d(x^{\ast\ast}, x): x\in X\}> 1-\epsilon.$$

(Here, *d* is the metric induced by the norm on \(X^{\ast\ast}\).)

- (b)
Let \(x^{\ast\ast}\) be as found in (a). Show that there exists \(x^\ast\in X^\ast\) with \(\|x^\ast\|=1\) and \(x^{\ast\ast}(x^\ast)>1-\epsilon/2\). Pick \(x\in X\) with \(\|x\|\leq 1\) such that \(x^\ast(x)>1-\epsilon/2\). Show that there exists \(y^\ast\in X^\ast\) with \(\|y^\ast\|=1\) and \(y^\ast(x^{\ast\ast}-x)>1-\epsilon\).

- (c)
Let *x* and \(x^\ast\) be as found in (b). Use Goldstine’s Theorem to show that there exists a \(y\in X\) with \(\|y\|\leq 1\) such that \(x^\ast(y)>1-\epsilon/2\) and \(y^\ast(y-x)>1-\epsilon\). Deduce that \(\|x+y\|>2-\epsilon\) and \(\|x-y\|>1-\epsilon\).

- (d)
Deduce that every uniformly convex space is reflexive.

**Exercise 7.27** For the following questions, assume

\(2<p<\infty\).

- (a)
Show that there is a constant \(c>0\) such that

$$\frac{1}{2}\Big(|1+t|^p + |1-t|^p\Big) \geq 1 + c^p \, |t|^p, t \in \mathbb{R}.$$

(*Hint:* Show that the function \(\frac{|1+t|^p + |1-t|^p - 2}{|t|^p}\) is bounded below.)

- (b)
Deduce from (a) that if *f* and *g* are functions in \(L_p(0,1)\), then

$$\frac{1}{2}\Big(\|f+g\|^p + \|f-g\|^p \Big) \geq \|f\|^p + c^p\, \|g\|^p.$$

- (c)
Conclude that \(L_p(0,1)\) is uniformly convex. (This is also true if \(1<p<2\), but it is a little more tricky to show.)