Consequences of Completeness

Abstract

The space $C[0,1]$ of continuous functions on the interval [0,1] can be equipped with many metrics. Two important examples are the metrics arising from the norms

$$\|f\|_\infty = \max_{s \in [0, 1]} |f(s)| \quad\mbox{and}\quad \|f\|_2 = \left(\int_0^1|f(s)|^2\, ds\right)^{1/2},$$

where $f\in C[0,1]$. The metric arising from the first norm is complete, whereas the metric induced by the second norm is not (i.e., there exist Cauchy sequences that fail to converge). Completeness of a metric is a very profitable property, as we shall see in this chapter. The first theorem we shall meet is a classical result about metric spaces called the Baire Category Theorem. It originated in Baire’s 1899 doctoral thesis, although metric spaces were not formally defined until later.

Exercises

Exercise 4.1 Let $\mathbb{R}$ be given the standard topology. Show that the closed set [0,1] is a $G_\delta$-set. Show that the open set $(0,1)$ is an $F_\sigma$-set.

Exercise 4.2 Show that the space $C[0,1]$ of continuous functions on the closed interval [0,1] is not complete in the norm

$$\|f\|_2=\left(\int_0^1 |f(s)|^2\, ds\right)^{1/2},\quad f\in C[0,1].$$

(The completion of $C[0,1]$ in the norm $\|\cdot\|_2$ is $L_2(0,1)$, by Lusin’s Theorem.)

Exercise 4.3 Let M and E be complete metric spaces. Suppose $h:M\rightarrow E$ is a homeomorphism onto its image (i.e., h is a continuous one-to-one map, and $h^{-1}|_{h(M)}$ is continuous). Show that h(M) is a $G_\delta$-set.

Exercise 4.4 Let X be a Banach space and suppose E is a dense linear subspace which is a $G_\delta$-set. Show that E = X.

Exercise 4.5 Show that if Y is a normed space which is homeomorphic to a complete metric space, then Y is a Banach space. (Hint: Consider Y as a dense subspace in its completion.)

Exercise 4.6 Let X and Y be Banach spaces and let $T:X\rightarrow Y$ be a bounded linear operator. If M is a closed subspace of X, show that either T(M) is first category in Y or $T(M)=Y$.

Exercise 4.7 Let $X=C^{(1)}[0,1]$ be the space of continuously differentiable functions on [0,1] and let $Y=C[0,1]$. Equip both spaces with the supremum norm $\|\cdot\|_\infty$. Define a linear map $T:X\rightarrow Y$ by $T(f) = f'$ for all functions $f\in C^{(1)}[0,1]$. Show that T has closed graph, but T is not continuous. Conclude that $\left(C^{(1)}[0,1], \|\cdot\|_\infty\right)$ is not a Banach space.

Exercise 4.8 Let $\phi \in C[0,1]$ be a function which is not identically 0. Show the set $M=\{\phi \, f: f\in C[0,1]\}$ is of the first category in $C[0,1]$ if and only if $\phi(x)=0$ for some $x\in [0,1]$.

Exercise 4.9 Let $\displaystyle (a_k)_{k\in\mathbb{Z}}$ be a sequence of complex scalars with only finitely many nonzero terms. Define a trigonometric polynomial $f:\mathbb{T}\rightarrow\mathbb{C}$ by $\displaystyle f(\theta) = \sum_{k\in\mathbb{Z}} a_k\, e^{ik\theta}.$ Show that $\hat{f}(n) = a_n$ for all $n\in\mathbb{Z}$.

Exercise 4.10 Show that there exists a function $f\in L_1(\mathbb{T})$ whose Fourier series fails to converge to f in the L ₁-norm. Precisely, show that if

$$S_N f = \sum_{k=-N}^N \hat{f}(k)\, e^{ik\theta},$$

then there exists an $f\in L_1(\mathbb{T})$ such that $\|f - S_Nf\|_{L_1(\mathbb{T})}$ does not tend to 0.

Exercise 4.11 Show that the Cesàro means $\frac{1}{N}(S_1 f + \cdots + S_N f)$ converge to f in the L ₁-norm for every $f\in L_1(\mathbb{T})$.

Exercise 4.12 Let X and Y be Banach spaces. If $T:X\rightarrow Y$ is a bijection, show that the adjoint map $T^\ast:Y^\ast\rightarrow X^\ast$ is also a bijection. Conclude that if T is an isomorphism of Banach spaces, then so is $T^\ast$.

Exercise 4.13 Show that the Hilbert transform of Example 4.40 is well-defined.

Exercise 4.14 Let $f:[1,\infty)\rightarrow\mathbb{R}$ be a continuous function. Suppose $(\xi_n)_{n=1}^\infty$ is a strictly increasing sequence of real numbers with $\xi_1\geq 1$, $\displaystyle\lim_{n\rightarrow\infty} \xi_n = \infty$, and $\displaystyle\lim_{n\rightarrow\infty} \frac{\xi_{n+1}}{\xi_n}=1$. If $\displaystyle\lim_{n\rightarrow\infty} f(\xi_n x) = 0$ for all $x\geq 1$, then prove that $\displaystyle\lim_{x\rightarrow\infty} f(x) = 0$.

Exercise 4.15 Show that $L_2(0,1)$ is of the first category in $L_1(0,1)$.

Exercise 4.16 Let $(\Omega,\mu)$ be a probability space and suppose there exists a sequence of disjoint sets $(E_n)_{n=1}^\infty$ such that $\mu(E_n)>0$ for all $n\in\mathbb{N}$. Show that $L_p(\Omega,\mu) \neq L_q(\Omega,\mu)$ if $1 \leq p < q < \infty$.

Exercise 4.17 Let X be an infinite-dimensional Banach space and suppose V is a closed subspace of X. If both V and $X/V$ are separable, then show X is separable.

Exercise 4.18 Identify the quotient space $c/c_0$.

Exercise 4.19 Let $1\leq p \leq \infty$ and let $V=\{(x_j)_{j=1}^\infty\in\ell_p: x_{2k}=0 \mbox{ for all} k\in\mathbb{N}\}$. Show that $\ell_p/V$ is isometrically isomorphic to ℓ _p .

Exercise 4.20 Find an example of a map $T:X\rightarrow Y$, where X and Y are normed spaces, such that T is a bounded linear bijection, but T ^-1 is not bounded. (Hint: X and Y cannot both be Banach spaces, or T ^-1 will be bounded by Corollary 4.30.)

Exercise 4.21 Show that the Closed Graph Theorem (Theorem 4.35) implies the Open Mapping Theorem (Theorem 4.29).