Separability of a vector space and its dual

Let’s recall that a topological space is separable when it contains a countable dense set. A link between separability and the dual space is following theorem:

Theorem: If the dual $$X^*$$ of a normed vector space $$X$$ is separable, then so is the space $$X$$ itself.

Proof outline: let $${f_n}$$ be a countable dense set in $$X^*$$ unit sphere $$S_*$$. For any $$n \in \mathbb{N}$$ one can find $$x_n$$ in $$X$$ unit ball such that $$f_n(x_n) \ge \frac{1}{2}$$. We claim that the countable set $$F = \mathrm{Span}_{\mathbb{Q}}(x_0,x_1,…)$$ is dense in $$X$$. If not, we would find $$x \in X \setminus \overline{F}$$ and according to Hahn-Banach theorem there would exist a linear functional $$f \in X^*$$ such that $$f_{\overline{F}} = 0$$ and $$\Vert f \Vert=1$$. But then for all $$n \in \mathbb{N}$$, $$\Vert f_n-f \Vert \ge \vert f_n(x_n)-f(x_n)\vert = \vert f(x_n) \vert \ge \frac{1}{2}$$. A contradiction since $${f_n}$$ is supposed to be dense in $$S_*$$.

We prove that the converse is not true, i.e. a dual space can be separable, while the space itself may be separable or not.

Introducing some normed vector spaces

Given a closed interval $$K \subset \mathbb{R}$$ and a set $$A \subset \mathbb{R}$$, we define the $$4$$ following spaces. The first three are endowed with the supremum norm, the last one with the $$\ell^1$$ norm.

• $$\mathcal{C}(K,\mathbb{R})$$, the space of continuous functions from $$K$$ to $$\mathbb{R}$$, is separable as the polynomial functions with coefficients in $$\mathbb{Q}$$ are dense and countable.
• $$\ell^{\infty}(A, \mathbb{R})$$ is the space of real bounded functions defined on $$A$$ with countable support.
• $$c_0(A, \mathbb{R}) \subset \ell^{\infty}(A, \mathbb{R})$$ is the subspace of elements of $$\ell^{\infty}(A)$$ going to $$0$$ at $$\infty$$.
• $$\ell^1(A, \mathbb{R})$$ is the space of summable functions on $$A$$: $$u \in \mathbb{R}^{A}$$ is in $$\ell^1(A, \mathbb{R})$$ iff $$\sum \limits_{a \in A} |u_x| < +\infty$$.

When $$A = \mathbb{N}$$, we find the usual sequence spaces. It should be noted that $$c_0(A, \mathbb{R})$$ and $$\ell^1(A, \mathbb{R})$$ are separable iff $$A$$ is countable (otherwise the subset $$\big\{x \mapsto 1_{\{a\}}(x),\ a \in A \big\}$$ is uncountable, and discrete), and that $$\ell^{\infty}(A, \mathbb{R})$$ is separable iff $$A$$ is finite (otherwise the subset $$\{0,1\}^A$$ is uncountable, and discrete).

Continue reading Separability of a vector space and its dual

A non complete normed vector space

Consider a real normed vector space $$V$$. $$V$$ is called complete if every Cauchy sequence in $$V$$ converges in $$V$$. A complete normed vector space is also called a Banach space.

A finite dimensional vector space is complete. This is a consequence of a theorem stating that all norms on finite dimensional vector spaces are equivalent.

There are many examples of Banach spaces with infinite dimension like $$(\ell_p, \Vert \cdot \Vert_p)$$ the space of real sequences endowed with the norm $$\displaystyle \Vert x \Vert_p = \left( \sum_{i=1}^\infty \vert x_i \vert^p \right)^{1/p}$$ for $$p \ge 1$$, the space $$(C(X), \Vert \cdot \Vert)$$ of real continuous functions on a compact Hausdorff space $$X$$ endowed with the norm $$\displaystyle \Vert f \Vert = \sup\limits_{x \in X} \vert f(x) \vert$$ or the Lebesgue space $$(L^1(\mathbb R), \Vert \cdot \Vert_1)$$ of Lebesgue real integrable functions endowed with the norm $$\displaystyle \Vert f \Vert = \int_{\mathbb R} \vert f(x) \vert \ dx$$.

Let’s give an example of a non complete normed vector space. Let $$(P, \Vert \cdot \Vert_\infty)$$ be the normed vector space of real polynomials endowed with the norm $$\displaystyle \Vert p \Vert_\infty = \sup\limits_{x \in [0,1]} \vert p(x) \vert$$. Consider the sequence of polynomials $$(p_n)$$ defined by
$p_n(x) = 1 + \frac{x}{2} + \frac{x^2}{4} + \cdots + \frac{x^n}{2^n} = \sum_{k=0}^{n} \frac{x^k}{2^k}.$ For $$m < n$$ and $$x \in [0,1]$$, we have $\vert p_n(x) - p_m(x) \vert = \left\vert \sum_{i=m+1}^n \frac{x^i}{2^i} \right\vert \le \sum_{i=m+1}^n \frac{1}{2^i} \le \frac{1}{2^m}$ which proves that $$(p_n)$$ is a Cauchy sequence. Also for $$x \in [0,1]$$ $\lim\limits_{n \to \infty} p_n(x) = p(x) \text{ where } p(x) = \frac{1}{1 - \frac{x}{2}}.$ As uniform converge implies pointwise convergence, if $$(p_n)$$ was convergent in $$P$$, it would be towards $$p$$. But $$p$$ is not a polynomial function as none of its $$n$$th-derivative always vanishes. Hence $$(p_n)$$ is a Cauchy sequence that doesn't converge in $$(P, \Vert \cdot \Vert_\infty)$$, proving as desired that this normed vector space is not complete. More generally, a normed vector space with countable dimension is never complete. This can be proven using Baire category theorem which states that a non-empty complete metric space is not the countable union of nowhere-dense closed sets.

Existence of a continuous function with divergent Fourier series

In that article, I provided an example of a continuous function with divergent Fourier series. We prove here the existence of such a function using Banach-Steinhaus theorem, also called uniform boundedness principle.

Theorem (Uniform Boundedness Theorem) Let $$(X, \Vert \cdot \Vert_X)$$ be a Banach space and $$(Y, \Vert \cdot \Vert_Y)$$ be a normed vector space. Suppose that $$F$$ is a set of continuous linear operators from $$X$$ to $$Y$$. If for all $$x \in X$$ one has $\sup\limits_{T \in F} \Vert T(x) \Vert_Y \lt \infty$ then $\sup\limits_{T \in F, \ \Vert x \Vert = 1} \Vert T(x) \Vert_Y \lt \infty$

Let’s take for $$X$$ the vector space $$\mathcal C_{2 \pi}$$ of continuous functions from $$\mathbb R$$ to $$\mathbb C$$ which are periodic with period $$2 \pi$$ endowed with the norm $$\Vert f \Vert_\infty = \sup\limits_{- \pi \le t \le \pi} \vert f(t) \vert$$. $$(\mathcal C_{2 \pi}, \Vert \cdot \Vert_\infty)$$ is a Banach space. For the vector space $$Y$$, we take the complex numbers $$\mathbb C$$ endowed with the modulus.

For $$n \in \mathbb N$$, the map $\begin{array}{l|rcl} \ell_n : & \mathcal C_{2 \pi} & \longrightarrow & \mathbb C \\ & f & \longmapsto & \displaystyle \sum_{p=-n}^n c_p(f) \end{array}$ is a linear operator, where for $$p \in \mathbb Z$$, $$c_p(f)$$ denotes the complex Fourier coefficient $c_p(f) = \frac{1}{2 \pi} \int_{- \pi}^{\pi} f(t) e^{-i p t} \ dt$

We now prove that
\begin{align*}
\Lambda_n &= \sup\limits_{f \in \mathcal C_{2 \pi}, \Vert f \Vert_\infty=1} \vert \ell_n(f) \vert\\
&= \frac{1}{2 \pi} \int_{- \pi}^{\pi} \left\vert \frac{\sin (2n+1)\frac{t}{2}}{\sin \frac{t}{2}} \right\vert \ dt = \frac{1}{2 \pi} \int_{- \pi}^{\pi} \left\vert h_n(t) \right\vert \ dt,
\end{align*} where one can notice that the function $\begin{array}{l|rcll} h_n : & [- \pi, \pi] & \longrightarrow & \mathbb C \\ & t & \longmapsto & \frac{\sin (2n+1)\frac{t}{2}}{\sin \frac{t}{2}} &\text{for } t \neq 0\\ & 0 & \longmapsto & 2n+1 \end{array}$ is continuous.
Continue reading Existence of a continuous function with divergent Fourier series

Counterexamples around Banach-Steinhaus theorem

In this article we look at what happens to Banach-Steinhaus theorem when the completness hypothesis is not fulfilled. One form of Banach-Steinhaus theorem is the following one.

Banach-Steinhaus Theorem
Let $$T_n : E \to F$$ be a sequence of continuous linear maps from a Banach space $$E$$ to a normed space $$F$$. If for all $$x \in E$$ the sequence $$T_n x$$ is convergent to $$Tx$$, then $$T$$ is a continuous linear map.

A sequence of continuous linear maps converging to an unbounded linear map

Let $$c_{00}$$ be the vector space of real sequences $$x=(x_n)$$ eventually vanishing, equipped with the norm $\Vert x \Vert = \sup_{n \in \mathbb N} \vert x_n \vert$ For $$n \in \mathbb N$$, $$T_n : E \to E$$ denotes the linear map defined by $T_n x = (x_1,2 x_2, \dots, n x_n,0,0, \dots).$ $$T_n$$ is continuous as for $$\Vert x \Vert \le 1$$, we have
\begin{align*}
\Vert T_n x \Vert &= \Vert (x_1,2 x_2, \dots, n x_n,0,0, \dots) \Vert\\
& = \sup_{1 \le k \le n} \vert k x_k \vert \le n \Vert x \Vert \le n
\end{align*} Continue reading Counterexamples around Banach-Steinhaus theorem

Counterexamples to Banach fixed-point theorem

Let $$(X,d)$$ be a metric space. Then a map $$T : X \to X$$ is called a contraction map if it exists $$0 \le k < 1$$ such that $d(T(x),T(y)) \le k d(x,y)$ for all $$x,y \in X$$. According to Banach fixed-point theorem, if $$(X,d)$$ is a complete metric space and $$T$$ a contraction map, then $$T$$ admits a fixed-point $$x^* \in X$$, i.e. $$T(x^*)=x^*$$.

We look here at counterexamples to the Banach fixed-point theorem when some hypothesis are not fulfilled.

First, let’s consider $\begin{array}{l|rcl} f : & \mathbb R & \longrightarrow & \mathbb R \\ & x & \longmapsto & x+1 \end{array}$ For all $$x,y \in \mathbb R$$ we have $$\vert f(x)-f(y) \vert = \vert x- y \vert$$. $$f$$ is not a contraction, but an isometry. Obviously, $$f$$ has no fixed-point.

We now prove that a map satisfying $d(g(x),g(y)) < d(x,y)$ might also not have a fixed-point. A counterexample is the following map $\begin{array}{l|rcl} g : & [0,+\infty) & \longrightarrow & [0,+\infty) \\ & x & \longmapsto & \sqrt{1+x^2} \end{array}$ Since $g^\prime(\xi) = \frac{\xi}{\sqrt{1+\xi^2}} < 1 \text{ for all } \xi \in [0, +\infty),$ by the mean value theorem $\vert g(x) - g(y)| = \vert g^\prime(\xi)\vert |x-y| < |x-y| \text{ for all } x, y \in [0, +\infty).$ However $$g$$ has no fixed-point. Finally, let's have a look to a space $$(X,d)$$ which is not complete. We take $$a,b \in \mathbb R$$ with $$0 < a < 1$$ and for $$(X,d)$$ the space $$X = \mathbb R \setminus \{\frac{b}{1-a}\}$$ equipped with absolute value distance. $$X$$ is not complete. Consider the map $\begin{array}{l|rcl} h : & X & \longrightarrow & X \\ & x & \longmapsto & ax + b \end{array}$ $$h$$ is well defined as for $$x \neq \frac{b}{1-a}$$, $$h(x) \neq \frac{b}{1-a}$$. $$h$$ is a contraction map as for $$x,y \in \mathbb R$$ $\vert h(x)-h(y) \vert = a \vert x - y \vert$ However, $$h$$ doesn't have a fixed-point in $$X$$ as $$\frac{b}{1-a}$$ is the only real for which $$h(x)=x$$.

Counterexample around Arzela-Ascoli theorem

Let’s recall Arzelà–Ascoli theorem:

Suppose that $$F$$ is a Banach space and $$E$$ a compact metric space. A subset $$\mathcal{H}$$ of the Banach space $$\mathcal{C}_F(E)$$ is relatively compact in the topology induced by the uniform norm if and only if it is equicontinuous and and for all $$x \in E$$, the set $$\mathcal{H}(x)=\{f(x) \ | \ f \in \mathcal{H}\}$$ is relatively compact.

We look here at what happens if we drop the requirement on space $$E$$ to be compact and provide a counterexample where the conclusion of Arzelà–Ascoli theorem doesn’t hold anymore.

We take for $$E$$ the real interval $$[0,+\infty)$$ and for all $$n \in \mathbb{N} \setminus \{0\}$$ the real function
$f_n(t)= \sin \sqrt{t+4 n^2 \pi^2}$ We prove that $$(f_n)$$ is equicontinuous, converges pointwise to $$0$$ but is not relatively compact.

According to the mean value theorem, for all $$x,y \in \mathbb{R}$$
$\vert \sin x – \sin y \vert \le \vert x – y \vert$ Hence for $$n \ge 1$$ and $$x,y \in [0,+\infty)$$
\begin{align*}
\vert f_n(x)-f_n(y) \vert &\le \vert \sqrt{x+4 n^2 \pi^2} -\sqrt{y+4 n^2 \pi^2} \vert \\
&= \frac{\vert x – y \vert}{\sqrt{x+4 n^2 \pi^2} +\sqrt{y+4 n^2 \pi^2}} \\
&\le \frac{\vert x – y \vert}{4 \pi}
\end{align*} using multiplication by the conjugate.

Which enables to prove that $$(f_n)$$ is equicontinuous.

We also have for $$n \ge 1$$ and $$x \in [0,+\infty)$$
\begin{align*}
\vert f_n(x) \vert &= \vert f_n(x) – f_n(0) \vert \le \vert \sqrt{x+4 n^2 \pi^2} -\sqrt{4 n^2 \pi^2} \vert \\
&= \frac{\vert x \vert}{\sqrt{x+4 n^2 \pi^2} +\sqrt{4 n^2 \pi^2}} \\
&\le \frac{\vert x \vert}{4 n \pi}
\end{align*}

Hence $$(f_n)$$ converges pointwise to $$0$$ and for $$t \in [0,+\infty), \mathcal{H}(t)=\{f_n(t) \ | \ n \in \mathbb{N} \setminus \{0\}\}$$ is relatively compact

Finally we prove that $$\mathcal{H}=\{f_n \ | \ n \in \mathbb{N} \setminus \{0\}\}$$ is not relatively compact. While $$(f_n)$$ converges pointwise to $$0$$, $$(f_n)$$ does not converge uniformly to $$f=0$$. Actually for $$n \ge 1$$ and $$t_n=\frac{\pi^2}{4} + 2n \pi^2$$ we have
$f_n(t_n)= \sin \sqrt{\frac{\pi^2}{4} + 2n \pi^2 +4 n^2 \pi^2}=\sin \sqrt{\left(\frac{\pi}{2} + 2 n \pi\right)^2}=1$ Consequently for all $$n \ge 1$$ $$\Vert f_n – f \Vert_\infty \ge 1$$. If $$\mathcal{H}$$ was relatively compact, $$(f_n)$$ would have a convergent subsequence with $$f=0$$ for limit. And that cannot be as for all $$n \ge 1$$ $$\Vert f_n – f \Vert_\infty \ge 1$$.

Distance between a point and a hyperplane not reached

Let’s investigate the following question: “Is the distance between a point and a hyperplane always reached?”

In order to provide answers to the question, we consider a normed vector space $$(E, \Vert \cdot \Vert)$$ and a hyperplane $$H$$ of $$E$$. $$H$$ is the kernel of a non-zero linear form. Namely, $$H=\{x \in E \text{ | } u(x)=0\}$$.

The case of finite dimensional vector spaces

When $$E$$ is of finite dimension, the distance $$d(a,H)=\inf\{\Vert h-a \Vert \text{ | } h \in H\}$$ between any point $$a \in E$$ and a hyperplane $$H$$ is reached at a point $$b \in H$$. The proof is rather simple. Consider a point $$c \in H$$. The set $$S = \{h \in H \text{ | } \Vert a- h \Vert \le \Vert a-c \Vert \}$$ is bounded as for $$h \in S$$ we have $$\Vert h \Vert \le \Vert a-c \Vert + \Vert a \Vert$$. $$S$$ is equal to $$D \cap H$$ where $$D$$ is the inverse image of the closed real segment $$[0,\Vert a-c \Vert]$$ by the continuous map $$f: x \mapsto \Vert a- x \Vert$$. Therefore $$D$$ is closed. $$H$$ is also closed as any linear subspace of a finite dimensional vector space. $$S$$ being the intersection of two closed subsets of $$E$$ is also closed. Hence $$S$$ is compact and the restriction of $$f$$ to $$S$$ reaches its infimum at some point $$b \in S \subset H$$ where $$d(a,H)=\Vert a-b \Vert$$. Continue reading Distance between a point and a hyperplane not reached

A solution of a differential equation not exploding in finite time

In this post, I mention that Peano existence theorem is valid for finite dimensional vector spaces, but not for Banach spaces of infinite dimension. I highlight here a second property of ordinary differential equations which is valid for finite dimensional vector spaces but not for infinite dimensional Banach spaces. Continue reading A solution of a differential equation not exploding in finite time

An empty intersection of nested closed convex subsets in a Banach space

We consider a decreasing sequence $$(C_n)_{n \in \mathbb{N}}$$ of non empty closed convex subsets of a Banach space $$E$$.

If the convex subsets are closed balls, their intersection is not empty. To see this let $$x_n$$ be the center and $$r_n > 0$$ the radius of the ball $$C_n$$. For $$0 \leq n < m$$ we have $$\Vert x_m-x_n\Vert \leq r_n – r_m$$ which proves that $$(x_n)_{n \in \mathbb{N}}$$ is a Cauchy sequence. As the space $$E$$ is Banach, $$(x_n)_{n \in \mathbb{N}}$$ converges to a limit $$x$$ and $$x \in \bigcap_{n=0}^{+\infty} C_n$$. Continue reading An empty intersection of nested closed convex subsets in a Banach space