**Exercise 3.1** Let *X* be a real inner product space with inner product \((\cdot,\cdot)\) and associated norm \(\|\cdot\|\). Prove the *Parallelogram Law:* If *x* and *y* are elements of *X*, then

$$\|x+y\|^2 + \|x-y\|^2 = 2\,(\|x\|^2+\|y\|^2).$$

**Exercise 3.2** Let *X* be a real inner product space with inner product \((\cdot,\cdot)\) and associated norm \(\|\cdot\|\). Verify the *polarization formula:* If *x* and *y* are in *X*, then

$$(x, y) = \frac{1}{4}\left( \|x+y\|^2 - \|x-y\|^2 \right).$$

**Exercise 3.3** Let *H* be a Hilbert space with norm \(\|\cdot\|\). Show that a bounded linear map \(S:H\rightarrow H\) is an orthogonal operator if and only if \(\|S^nx\|=\|x\|\) for all \(n\in\mathbb{N}\) and \(x\in H\). (*Hint:* Use Exercise 3.2.)

**Exercise 3.4** Let *X* and *Y* be normed vector spaces. If *x* is a nonzero vector in *X*, and \(y\in Y\), show there exists a bounded linear map \(T:X\rightarrow Y\) such that \(T(x)=y\).

**Exercise 3.5** In this exercise, we complete the proof of Theorem 3.32. Let

*X* be a Banach space with bidual

\(X^{\ast\ast}\) and let

\(j:X\rightarrow X^{\ast\ast}\) be the natural embedding.

- (a)
Show that *j* is an injective bounded linear map.

- (b)
Show that *j*(*X*) is a closed subset of \(X^{\ast\ast}\). (You may wish to use Exercise 1.10.)

**Exercise 3.6** Let *X* and *Y* be normed spaces. Show that if \(\mathcal{L}(X,Y)\) is a Banach space, then *Y* must be a Banach space. (This is the converse to Proposition 1.11.)

**Exercise 3.7** Let *X* and *Z* be Banach spaces and let \(T\in\mathcal{L}(X,Z)\). Show that *T* is an open map if and only if *T* maps the open unit ball of *X* to an open set in *Z*.

**Exercise 3.8** Prove Proposition 3.40.

**Exercise 3.9** Let *p* be a sublinear functional on a real vector space *V*. Show that

$$p(x) = \max\{f(x):f\leq p, \, f \mbox{ linear}\},\quad x\in V.$$

Conversely, show that a functional *q* of the form

$$q(x) = \sup_{f\in A} f(x),\quad x\in V,$$

where *A* is some collection of linear functionals, is necessarily sublinear.

**Exercise 3.10** **(Fekete’s Lemma [10])** Let \((a_n)_{n=1}^\infty\) be a sequence of real numbers such that

$$a_{m+n}\leq a_m + a_n, \quad\{m,n\} \subseteq \mathbb{N}.$$

Show that if the sequence \((a_n/n)_{n=1}^\infty\) is bounded below, then

$$\lim_{n\rightarrow\infty}\frac{a_n}{n} = \inf_{n\in\mathbb{N}}\frac{a_n}{n}.$$

(*Hint:* For any \(m\in\mathbb{N}\), show that \(\displaystyle \limsup_{n\rightarrow\infty} \frac{a_n}{n} \leq \frac{a_m}{m}\) by writing \(n=k\,m+r\), where \(\{k,r\}\subseteq\mathbb{N}\) and \(0\leq r \leq m-1\).)

**Exercise 3.11** Let *V* be a real vector space and suppose *p* is a sublinear functional on *V*. Suppose \(T:V\rightarrow V\) is a linear map such that \(p(Tx) = p(x)\). Show, using Exercise 3.10, that

$$q(x)=\lim_{n\rightarrow\infty}\frac{1}{n}\, p(x+Tx+\cdots+T^{n-1}x),\quad x\in V,$$

defines a sublinear functional with \(q\leq p\). Show further that if *f* is a linear functional with \(f\leq p\), then *f* is \(T\) *-invariant* (i.e., \(f(Tx)=f(x)\) for all \(x\in V\)) if and only if \(f\leq q\).

**Exercise 3.12.** Show that a linear functional *L* on \(\ell_\infty\) is a Banach limit if and only if

$$L(\xi)\leq \lim_{n\rightarrow\infty}\sup_{k\in\mathbb{N}}\frac{\xi_{k+1}+\cdots+\xi_{k+n}}{n},\quad \xi = (\xi_j)_{j=1}^\infty\in\ell_\infty.$$

**Exercise 3.13** A sequence

\(\xi\in\ell_\infty\) is called

*almost convergent* to

*α* if

\(L(\xi)=\alpha\) for all Banach limits

*L*.

- (a)
Show that ξ is almost convergent to *α* if and only if

$$\lim_{n\rightarrow\infty}\sup_{k\in\mathbb{N}}\Big|\frac{\xi_{k+1}+\cdots+\xi_{k+n}}{n}-\alpha\Big| = 0.$$

- (b)
Show that for any *θ*, the sequence \(\left(\sin (n\theta)\right)_{n=1}^\infty\) is almost convergent to 0.

**Exercise 3.14** If \(x=(x_j)_{j=1}^\infty\) and \(y=(y_j)_{j=1}^\infty\) are sequences in \(\ell_\infty\), let *xy* be the sequence \((x_j\, y_j)_{j=1}^\infty\). Show for any Banach limit *L*, there are sequences *x* and *y* in \(\ell_\infty\) such that \(L(xy) \neq L(x)\, L(y)\). (Notice that \(L(xy) = L(x)\, L(y)\) if *x* and *y* are in *c*.)

**Exercise 3.15** Prove that \(\langle \cdot, \cdot \rangle\) is an inner product on *H* in the proof of Proposition 3.21.

**Exercise 3.16** Show that a bounded linear functional \(f:c_0\rightarrow\mathbb{R}\) has a *unique* Hahn–Banach extension \(\tilde{f}:\ell_\infty\rightarrow\mathbb{R}\).

**Exercise 3.17** Let \(\displaystyle E=\{(x_n)_{n=1}^\infty\in\ell_1: x_{2k-1} = 0 \mbox{ for all} k\in\mathbb{N}\}\). Show that *E* is a closed subspace of *ℓ* _{1}. Prove that any nonzero bounded linear functional on *E* has more than one Hahn–Banach extension to *ℓ* _{1}.

**Exercise 3.18** Let \(\displaystyle E=\left\{f\in L_2(0,1): \int_0^1 x f(x)\, dx=0\right\}\). Define a bounded linear functional *Λ* on *E* by

$$\Lambda(f) = \int_0^1 x^2 f(x)\, dx,\quad f\in E.$$

Find the (unique) Hahn–Banach extension of *Λ* to \(L_2(0,1)\) and determine \(\|\Lambda\|\).

(*Hint:* Use the fact that \(\displaystyle \int_0^1 x^2f(x)\, dx = \int_0^1 (x^2 + ax) f(x)\, dx\) on *E*, for all \(a\in\mathbb{R}\).)