**Exercise 2.1** Show that *c* _{0} is a closed subspace of *c* and that *c* is a closed subspace of \(\ell_\infty\).

**Exercise 2.2** Show that the dual of *c* _{0} can be identified with *ℓ* _{1}, and that the dual of *ℓ* _{1} can be identified with \(\ell_\infty\).

**Exercise 2.3** Let \((\Omega,\Sigma,\mu)\) be a positive finite measure space. Show that \(L_1(\Omega,\mu)^\ast\) can be identified with \(L_\infty(\Omega,\mu)\). (This completes the proof of Theorem 2.13.)

**Exercise 2.4** Let \(g\in L_1(0,1)\), the Banach space of *L* _{1}-functions on [0,1] with Lebesgue measure, and define a measure on [0,1] by \(\nu(A)=\int_A g(t)\, {\it dt}\), where *A* is a Borel subset of [0,1]. Show that \(|\nu|(A) = \int_A |g(t)|\, {\it dt}\) for all Borel subsets *A* of [0,1]. (See Example 2.17(c) and the comments following it.)

**Exercise 2.5** Let *K* be a compact metric space. Prove that *C*(*K*) is a Banach space when given the supremum norm. (That is, prove Theorem 2.27.)

**Exercise 2.6** Let *δ* _{0} denote the linear functional on \(C[0,1]\) given by evaluation at 0. That is, \(\delta_0(f)=f(0)\) for all \(f\in C[0,1]\). Show that *δ* _{0} is bounded on \(C[0,1]\) when equipped with the \(\|\cdot\|_\infty\)-norm, but not when equipped with the \(\|\cdot\|_1\)-norm.

**Exercise 2.7** Use the theorems of Sect. 2.3 to prove that *ℓ* _{ p } is complete in the *p*-norm for \(1\leq p \leq \infty\). (You may assume the theorems of Sect. 2.3 remain true for σ-finite measure spaces.)

**Exercise 2.8** Verify that any Cauchy sequence in *c* _{0} (equipped with the supremum norm) converges to a limit in *c* _{0}. Conclude that *c* _{0} is a Banach space.

**Exercise 2.9** Prove that *c* _{0} is not a Banach space in the \(\|\cdot\|_2\)-norm.

**Exercise 2.10** Let

\(1\leq p < q \leq \infty\).

- (a)
Denote by \(\ell_p^n\) the finite-dimensional vector space \(\mathbb{R}^n\) equipped with the norm

$$\|(x_1,\ldots,x_n)\|_p=\left(|x_1|^p + \cdots + |x_n|^p\right)^{1/p}.$$

Show that the norms \(\|\cdot\|_p\) and \(\|\cdot\|_q\) are equivalent on \(\mathbb{R}^n\).

- (b)
Show that \(\ell_p \subseteq\ell_q\), but *ℓ* _{ q } is not a subset of *ℓ* _{ p }.

**Exercise 2.11** Let \(x\in\ell_{r}\) for some \(r<\infty\). Show that \(x\in\ell_p\) for all \(p\geq r\) and prove that \(\|x\|_p \rightarrow \|x\|_\infty\) as \(p\rightarrow\infty\).

**Exercise 2.12** Suppose

\((\Omega, \mu)\) is a positive measure space and let

\(1\leq p<q \leq \infty\).

- (a)
Prove that if \(\mu(\Omega)<\infty\), then \(\|f\|_p \leq C_{p,q}\, \|f\|_q\) for all measurable functions *f*, where \(C_{p,q}\) is a constant that depends on *p* and *q*.

- (b)
Show that the assumption \(\mu(\Omega)<\infty\) cannot be omitted in (a).

- (c)
Find a real-valued function *f* on [0,1] such that \(\|f\|_p<\infty\) but \(\|f\|_q=\infty\).

**Exercise 2.13** Suppose

\((\Omega, \mu)\) is a positive measure space such that

\(\mu(\Omega)=1\).

- (a)
If \(1\leq p<q \leq \infty\), then show \(\|f\|_p \leq \|f\|_q\) for all measurable functions *f*. (See Exercise 2.12.)

- (b)
Assume that *f* is an essentially bounded measurable function and prove that \(\|f\|_p\rightarrow\|f\|_\infty\) as \(p\rightarrow\infty\).

**Exercise 2.14** Let

\(\ell_p^2\) be the finite-dimensional vector space

\(\mathbb{R}^2\) equipped with the

\(\|\cdot\|_p\)-norm for

\(1\leq p \leq \infty\).

- (a)
What do the closed unit balls \(B_{\ell_1^2}\), \(B_{\ell_2^2}\), and \(B_{\ell_\infty^2}\) represent geometrically?

- (b)
Let *a* and *b* be nonzero real numbers and define a function on \(\mathbb{R}^2\) by

$$\|(x,y)\|_E=\left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right)^{1/2}, \quad (x,y)\in \mathbb{R}^2.$$

Prove that \(\|\cdot\|_E\) is a norm on \(\mathbb{R}^2\) and identify geometrically the closed unit ball in \((\mathbb{R}^2, \|\cdot\|_E)\).

**Exercise 2.15** Let

*M* be a metric space with subset

*E*. A set

*V* is said to be

*open in the subspace topology on* \(E\) if there exists a set

*U* that is open in

*M* and

\(V=U\cap E\).

- (a)
Show that a closed subset of a complete metric space is a complete metric space.

- (b)
Show that \(\mathbb{N}\), the set of natural numbers, is a locally compact metric space with the metric \(d(x,y)=|x-y|\) for all *x* and *y* in \(\mathbb{N}\). Conclude that the one-point compactification \(\mathbb{N}\cup\{\infty\}\) of the natural numbers is a compact metric space. (See Appendix B.1 for the definition of the one-point compactification.)

- (c)
Show that the one-point compactification of the natural numbers \(\mathbb{N}\cup\{\infty\}\) (from part (b)) is homeomorphic to \(\{1/n:n\in\mathbb{N}\}\cup\{0\}\), where the latter set is given the subspace topology inherited from \(\mathbb{R}\).

- (d)
Conclude that *c* is a Banach space. (See Example 2.22.)

**Exercise 2.16** Consider the interval [0,1]. From this set, remove the open subinterval \((\frac{1}{3},\frac{2}{3})\), the so-called *middle third*. This leaves the union of two closed intervals: \([0,\frac{1}{3}]\cup[\frac{2}{3},1]\). From each of these, again remove the middle third. What remains is the union of four closed intervals: \([0,\frac{1}{9}]\cup[\frac{2}{9},\frac{1}{3}]\cup[\frac{2}{3},\frac{7}{9}]\cup[\frac{8}{9},1]\). Once again, from each remaining set remove the middle third. Continue this process indefinitely to create *Cantor’s Middle Thirds Set*. This set, which we denote \(\mathcal{K}\), can be written explicitly as follows:

$$\mathcal{K} = [0,1] \setminus \bigcup_{n=1}^\infty \bigcup_{k=0}^{3^{n-1}-1} \left( \frac{3k+1}{3^n}, \frac{3k+2}{3^n}\right).$$

- (a)
Show that \(\mathcal{K}\) coincides with the Cantor set \(\mathcal{C}\) of Example 2.21. (*Hint:* You may wish to use the fact that every nonzero number has a unique nonterminating ternary expansion.)

- (b)
Evidently \(\mathcal{K}\) is not empty because it contains the endpoints of the middle third sets. Show that \(\mathcal{K}\) contains other numbers by showing that \(\frac{1}{4}\in\mathcal{K}\). (*Hint:* Consider the geometric series \(\sum_{j=1}^\infty 3^{-kj}\), where \(k\in\mathbb{N}\).)

- (c)
Show that \(\mathcal{K}\) is uncountable. (*Hint:* Use diagonalization and \(\mathcal{K}=\mathcal{C}\).)

- (d)
Let *m* be Lebesgue measure on [0,1]. Show that \(m(\mathcal{K})=0\).

- (e)
Let *μ* _{ G } be the Cantor measure from Example 2.21. Show that \(\mu_G(\mathcal{K})=1\).

**Exercise 2.17** Let \(\mathcal{K}\) be Cantor’s Middle Thirds Set from Exercise 2.16. After the first middle third is removed, two closed intervals remain. Call these two sets \(E_{1,1}\) and \(E_{1,2}\). After the middle thirds are removed from \(E_{1,1}\) and \(E_{1,2}\), there will remain four closed intervals. Label these sets \(E_{2,1}, E_{2,2}, E_{2,3}, E_{2,4}\), ordering them from left to right, as they appear on the unit interval. After the process has been repeated *n* times, there will remain \(2^n\) closed intervals, each of length \(\frac{1}{3^n}\). Label these sets \(E_{n,1}, \ldots, E_{n,2^n}\), again from left to right, as they appear on the unit interval, so that \(x_1\in E_{n,k_1}\) and \(x_2\in E_{n,k_2}\) implies that \(x_1<x_2\) whenever \(k_1<k_2\). Observe that

$$\mathcal{K}=\bigcap_{n=1}^\infty \bigcup_{k=1}^{2^n} E_{n,k}.$$

For each \(n\in\mathbb{N}\) and each \(k\in\{1,\ldots,2^n\}\), let \(E_{n,k}=[a_{n,k}, b_{n,k}]\) and define a function \(G_n:[0,1]\rightarrow[0,1]\) as follows:

$$G_n(x) =\begin{cases} \frac{b_{n,k}-x}{3^{-n}}\frac{k-1}{2^n}+ \frac{x-a_{n,k}}{3^{-n}}\frac{k}{2^n} & \mbox{if}\; a_{n,k} \leq x \leq b_{n,k}\ \mbox{ for}\;\, k\in\{1, \ldots, 2^n\}, \\ \frac{k}{2^n} & \mbox{if}\; b_{n,k}<x<a_{n,k+1}\ \mbox{for} \;\, k\in\{1,\ldots,2^n-1\}.\end{cases}$$

show that \((G_n)_{n=1^{\infty}}\) converges uniformly to *G*, the cantor function (from Example 2.21), and deduce that *G* is continuous on [0, 1].