Ngô Quốc Anh

April 14, 2019

Extending functions between metric spaces: Continuity, uniform continuity, and uniform equicontinuity

Filed under: Giải Tích 3, Giải tích 8 (MA5206) — Tags: — Ngô Quốc Anh @ 15:02

This topic concerns a very classical question: extend of a function f : X \to Y between two metric spaces to obtain a new function \widetilde f : \overline X \to Y enjoying certain properties. I am interested in the following three properties:

  • Continuity,
  • Uniformly continuity,
  • Pointwise equi-continuity, and
  • Uniformly equi-continuity.

Throughout this topic, by X and Y we mean metric spaces with metrics d_X and d_Y respectively.

CONTINUITY IS NOT ENOUGH. Let us consider the first situation where the given function f : X \to Y is only assumed to be continuous. In this scenario, there is no hope that we can extend such a continuous function f to obtain a new continuous function \widetilde f : \overline X \to Y. The following counter-example demonstrates this:

Let X = [0,\frac 12 ) \cup (\frac 12, 1] and let f be any continuous function on X such that there is a positive gap between f(\frac 12+) and f(\frac12-). For example, we can choose

\displaystyle f(x)=\begin{cases}x^2&\text{ if } x<\frac 12,\\x^3 & \text{ if } x>\frac 12.\end{cases}

Since f is monotone increasing, we clearly have

\displaystyle f(\frac12-)-f(\frac 12+)=\frac18.

Hence any extension \widetilde f of f cannot be continuous because \widetilde f will be discontinuous at x =\frac 12. Thus, we have just shown that continuity is not enough. For this reason, we require f to be uniformly continuous.

SIMPLE OBSERVATIONS. We start with the following basic results.

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February 23, 2017

In a normed space, finite linearly independent systems are stable under small perturbations

Filed under: Giải tích 8 (MA5206) — Ngô Quốc Anh @ 23:21

In this topic, we show that in a normed space, any finite linearly independent system is stable under small perturbations. To be exact, here is the statement.

Suppose (X, \|\cdot\|) is a normed space and \{x_1,...,x_n\} is a set of linearly independent elements in X. Then \{x_1,...,x_n\} is stable under a small perturbation in the sense that there exists some small number \varepsilon>0 such that for any \|y_i\| < \varepsilon with 1 \leqslant i \leqslant n, the all elements of \{x_1+y_1,...,x_n+y_n\} are also linearly independent.

We prove this result by way of contradiction. Indeed, for any \varepsilon>0, there exist n elements y_i \in X with \|y_i\| < \varepsilon such that all elements of \{x_1+y_1,...,x_n+y_n\} are linearly dependent, that is, there exist real numbers \alpha_i with 1 \leqslant i \leqslant n such that

\displaystyle \alpha_1 (x_1+y_1) + \cdots + \alpha_n (x_n+y_n) =0

with

\displaystyle |\alpha_1| + \cdots + |\alpha_n| >0.

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May 22, 2010

The third and fouth fundamental results in the calculus of variation

Filed under: Giải tích 8 (MA5206), PDEs — Ngô Quốc Anh @ 17:56

Followed by this entry where the following questions have been discussed

  1. Boundedness implies weakly convergence: If E is a reflexive Banach space and \{x_n\}_n \subset E is a bounded sequence. Then up to a subsequence x_n converges weakly to some x in X.
  2. Weakly convergence becomes strongly convergence via compact operator: A compact operator C : E \to X between Banach spaces maps every weakly convergent sequence in E into one that converges strongly in X.

Now I shall discuss more results which appear frequently in the calculus of variation.

Let us recall over a minifold M, the Sobolev norm H^1(M) (or W^{1,2}(M)) is defined by

\displaystyle \|u\|_{H^1}^2=\|u\|_{L^2}^2+\|\nabla u\|_{L^2}^2.

It is immediately to see that if u_n \to u strongly in H^1, i.e. \|u_n-u\|_{H^1}\to 0 then u_n \to u strongly in L^2 and \nabla u_n \to \nabla u strongly in L^2.

It turns out to discuss what happen to weakly convergence. Actually, we shall prove the following important result, called the third fundamental result.

Weakly convergence in Sobolev spaces implies weakly convergence in L^p spaces. We assume u_n \rightharpoonup u in H^1. We shall prove both u_n and \nabla u_n converge weakly to u and \nabla u in L^2, respectively.

By the principle of uniform boundedness, any weakly convergence sequence is bounded in the norm. Consequently, \{u_n\}_n and \{\nabla u_n\}_n are bounded in L^2. By the weak compactness of balls in L^2, there is a subsequence n_k such that

\displaystyle u_{n_k} \rightharpoonup v, \quad \nabla u_{n_k} \rightharpoonup w

in L^2 (i.e., both v, w are in L^2). Since the weak convergence in L^2 implies the convergence in \mathcal D' the dual space of \mathcal D-the space of test functions. It follows that w = \nabla v and, hence, v \in H^1. It follows that u\equiv v and thus

\displaystyle u_{n_k} \rightharpoonup u, \quad \nabla u_{n_k} \rightharpoonup \nabla u

in L^2 as desired.

Now we consider the reverse case. We shall prove the following

Weakly convergence in L^p spaces plus the boundedness implies weakly convergence in Sobolev spaces. We assume u_n \rightharpoonup u \in L^2 in L^2 and \|u_n\|_{H^1} is bounded. We shall prove that u \in H^1 and u_n \rightharpoonup u in H^1.

Since \{u_n\} is bounded in H^1, by the first fundamental result, u_n \rightharpoonup v in H^1 for some v \in H^1. By the third fundamental result above, u_n \rightharpoonup v in L^2. It follows from the uniqueness of weak limit that u \equiv v which implies u \in H^1.

In order to prove u_n \rightharpoonup u in H^1, we shall use the following result whose proof is based on the simple contradiction argument.

Let X be a topological space. A sequence \{x_n\} \subset X  converges to x \in X (in the topological of X) if and only if any subsequence of \{x_n\} contains a sub-subsequence that converges to x.

Let us pick a particular subsequence of \{u_n\} and rename it back to \{u_n\} for simplicity. It suffices to prove that \{u_n\} contains a subsequence that converges to u weakly in H^1. Followed the proof of the third fundamental result, there is a subsequence of \{u_n\} and a function v \in H^1 such that

\displaystyle u_{n_k} \rightharpoonup v, \quad \nabla u_{n_k} \rightharpoonup v

both in L^2. It follows from the definition of weakly convergence in H^1 that in fact we get

\displaystyle u_{n_k} \rightharpoonup v

in H^1. The reason is the following:

\displaystyle (u_{n_k},\varphi)_{L^2}+(\nabla u_{n_k},\nabla\varphi)_{L^2} \to (v,\varphi)_{L^2} + (\nabla v, \varphi)_{L^2}

for any \varphi \in H^1. Having this and the fact that weak limit is unique we deduce that u \equiv v. The latter now implies

u_{n_k} \rightharpoonup u

in H^1. The proof follows.

See also: Two fundamental results in the calculus of variation

May 15, 2010

Two fundamental results in the calculus of variation

Filed under: Giải tích 8 (MA5206), PDEs — Ngô Quốc Anh @ 20:28

I suddenly think that I should post this entry ‘cos sometimes I don’t remember these stuffs. These results appear frequently in solving PDEs especially when using the direct method. For example, the simplest case is the following eigenvalue problem

\displaystyle -{\rm div}(|\nabla u|^{p-2}\nabla u)=\lambda |u|^{p-2}u

over a bounded domain \Omega \subset \mathbb R^n with Dirichlet boundary condition. We assume p<n. Our aim is to show the existence of the first eigenvalue \lambda_1>0. Obviously, our problem is to solve the following optimization

\displaystyle\mathop {\inf }\limits_{u \in W^{1,2}_0(\Omega)} \left\{ {\int_\Omega {|\nabla u{|^p}dx} :\int_\Omega {|u{|^p}dx} = 1} \right\}.

The direct method says that we firstly select a minimizing sequence, say \{u_n\}_n, then we need to prove \{u_n\} is convergent. There are two steps in the above argument which lead to this entry. Our first claim is the following.

Boundedness implies weakly convergence. The first result says that

If E is a reflexive Banach space and \{x_n\}_n \subset E is a bounded sequence. Then up to a subsequence x_n converges weakly to some x in X.

The proof of this claim can be found in a book due to Brezis (Theorem III.27). Interestingly, its converse also holds by the Eberlein-Šmulian theorem.

Theorem (Eberlein-Šmulian). Suppose E is a Banach space such that every bounded sequence \{x_n\}_n contains a weakly convergent subsequence. Then E is reflexive.

There was an elementary proof of this theorem. We refer the reader to a paper due to Whitley [here]. Let us get back to our optimization problem. Once we have a minimizing sequence \{u_n\}_n \subset W^{1,p}_0(\Omega) it is clear to see that \{u_n\}_n is bounded in W^{1,p}_0(\Omega) since

\displaystyle\int_\Omega {|{u_n}{|^p}dx} = 1

and

\displaystyle\int_\Omega {|\nabla {u_n}{|^p}dx} \leqslant C, \quad \forall n.

By using the first claim, u_n converges weakly to some u \in W^{1,p}_0(\Omega). It is worth noticing that by saying u_n \rightharpoonup u in W^{1,p}_0(\Omega) we mean u_n \rightharpoonup u in L^p(\Omega) and Du_n \rightharpoonup Du in L^p(\Omega, \mathbb R^n). Now we need further argument

Weakly convergence becomes strongly convergence via compact operator. This second result says that

A compact operator C : E \to X between Banach spaces maps every weakly convergent sequence in E into one that converges strongly in X.

The proof of this relies on the contradiction argument and the fact that once a sequence converges strongly to some limit, this limit is unique. However, the converse is no long true. For example, by the Schur theorem, a sequence \{x_n\} in \ell^1 converges weakly, it also converges strongly. We take C to be the identity I in \ell^1. Since \ell^1 has infinity dimensional, by using the Riesz theorem, I cannot be compact.

Using this claim we deduce that u_n converges strongly to u in L^q for any 1\leqslant q<p^\star=\frac{np}{n-p}. By using the Minkowski and Holder inequalities we can show that u satisfies the constraint. It now follows from the weakly lower semi-continuous of norm that u indeed satisfies the equation. The proof follows.

May 14, 2010

Symmetrization: Schwarz symmetrization

Filed under: Giải tích 8 (MA5206) — Tags: — Ngô Quốc Anh @ 15:34

Given a measurable subset E \subset \mathbb R^N, we denote its N-dimensional Lebesgue measure by |E|. We will denote by E^\star the open ball centered at the origin and having the same measure as E, i.e. |E^\star|=|E|. The norm of vector x \in \mathbb R^n will be denoted by |x|. Finally, we will denote by \omega_N the volume of the unit ball in \mathbb R^N. It is worth recalling that

\displaystyle \omega_N=\frac{\pi^\frac{N}{2}}{\Gamma \left(\frac{N}{2}+1\right)}

where \Gamma us the usual gamma function.

Definition (Schwarz symmetrization). Let \Omega \subset \mathbb R^N be a bounded domain. Let u : \Omega \to \mathbb R be a measurable function. Then, its Schwarz symmetrization (or the spherically symmetric and decreasing rearrangement) is the function u^\star : \Omega^\star \to \mathbb R defined by

u^\star(x)=u^\sharp (\omega_N|x|^N), \quad x \in \Omega^\star.

Observe that if R is the radius of \Omega^\star, then

\displaystyle\begin{gathered} \int_{{\Omega ^ \star }} {{u^ \star }(x)dx} = \int_{{\Omega ^ \star }} {{u^\sharp }({\omega _N}{{\left| x \right|}^N})dx} \hfill \\ \qquad= \int_0^R {{u^\sharp }({\omega _N}{{\left| x \right|}^N})N{\omega _N}{\tau ^{N - 1}}d\tau } \hfill \\ \qquad= \int_0^{|{\Omega ^ \star }|} {{u^\sharp }(s)ds} \hfill \\ \qquad= \int_0^{|\Omega |} {{u^\sharp }(s)ds} . \hfill \\ \end{gathered}

We obviously have the following properties of Schwarz symmetrization (more…)

May 11, 2010

Compact embedding of Hölder spaces

Filed under: Giải tích 8 (MA5206) — Tags: — Ngô Quốc Anh @ 2:51

Hölder continuous. In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, \alpha, such that

\displaystyle | f(x) - f(y) | \leqslant C |x - y|^{\alpha}

for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number \alpha is called the exponent of the Hölder condition. If \alpha = 1, then the function satisfies a Lipschitz condition. If \alpha = 0, then the function simply is bounded.

Hölder spaces. Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations, and in dynamical systems. The Hölder space C^{k,\alpha}(\Omega), where \Omega is an open subset of some Euclidean space and k \geqslant 0 an integer, consists of those functions on \Omega having derivatives  up to order k and such that the k-th partial derivatives are Hölder continuous with exponent \alpha, where 0 <\alpha \leqslant 1. This is a locally convex topological vector space.

If the Hölder coefficient

\displaystyle | f |_{C^{0,\alpha}} = \sup_{x,y \in \Omega} \frac{| f(x) - f(y) |}{|x-y|^\alpha},

is finite, then the function f is said to be (uniformly) Hölder continuous with exponent \alpha in \Omega. In this case, Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of \Omega, then the function f is said to be locally Hölder continuous with exponent \alpha in \Omega.

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May 2, 2010

Symmetrization: The Decreasing Rearrangement

Filed under: Giải tích 8 (MA5206), PDEs — Tags: , , — Ngô Quốc Anh @ 15:37

Given a measurable subset E \subset \mathbb R^N, we denote its N-dimensional Lebesgue measure by |E|.

Let \Omega be a bounded measurable set. Let u :\Omega \to \mathbb R be a measurable function. For t \in \mathbb R, the level set \{u>t\} is defined as

\displaystyle \{u>t\}=\{x\in \Omega: u(x)>t\}.

The sets \{u<t\}, \{u \geqslant t\}, \{u=t\} and so on are defined by analogy. Then the distribution function of u is given by

\displaystyle \mu_u(t)=|\{u>t\}|.

This function is a monotonically decreasing function of t and for t \geq {\rm esssup}(u) we have \mu_u(t)=0 while for t\leqslant {\rm essinf}(u), we have \mu_u(t)=|\Omega|. Thus the range of \mu_u is the interval [0, |\Omega|].

Definition (Decreasing rearrangement). Let \Omega \subset \mathbb R^N be bounded and let u :\Omega \to \mathbb R be a measurable function. Then the (unidimensional) decreasing rearrangement of u, denoted by u^\sharp, is defined on [0, |\Omega|] by

\displaystyle {u^\sharp }(s) = \begin{cases} {\rm esssup} (u),& s = 0, \hfill \\ \mathop {\inf }\limits_t \left\{ {t:{\mu _u}(t) < s} \right\}, & s > 0. \hfill \\ \end{cases}

Essentially, u^\sharp is just the inverse function of the distribution function \mu_u of u. The following properties of the decreasing rearrangement are immediate from its definition.

Proposition 1. Let u : \Omega \to \mathbb R^N where \Omega \subset \mathbb R^N is bounded. Then u^\sharp is a nonincreasing and left-continuous function.

Proposition 2. The mapping u \mapsto u^\sharp is non-decreasing, i.e. if u\leqslant v in the sense that u(x) \leqslant v(x) for all x, where u and v are real-valued functions on \Omega then u^\sharp \leqslant v^\sharp.

We now see that u^\sharp is indeed a rearrangement of u.

Proposition 3. The function u : \Omega \to \mathbb R and u^\sharp : [0,|\Omega|] \to \mathbb R are equimeasurable (i.e. they have the same distribution function), i.e. for all t

\displaystyle |\{u >t\}|=|\{u^\sharp >t\}|.

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April 17, 2010

A Theorem of Banach and Saks

Filed under: Giải tích 8 (MA5206) — Ngô Quốc Anh @ 23:39

According to Banach and Saks, every bounded sequence in L^p or \ell^p (1<p< \infty) has a subsequence whose Cesaro-means converge strongly. More generally, every uniformly convex Banach space possesses this so-called Banach-Saks property, as shown by Kakutani. In particular, every Hilbert space has this property. In nonlinear analysis, by utilizing a duality mapping some assertions which are valid in the case of Hilbert spaces are extended to the case of special classes of Banach spaces. Especially in the case of Banach spaces with a uniformly convex conjugate space, such extentions are often obtained since a duality mapping is uniformly strongly continuous on each bounded subset of such a Banach space.

The Banach-Saks theorem in L^2 states that

Theorem (Banach-Saks for L^2 spaces). Given in L^2 a sequence \{f_n\}_n which converges weakly to an element f, we can select a subsequence \{f_{n_k}\}_k such that the arithmetic means

\displaystyle\frac{{{f_{{n_1}}} + {f_{{n_2}}} + \cdots + {f_{{n_k}}}}}{k}

converge in strongly to f.

This theorem is due to the two Polish geometers S.  Banach and S. Saks, whose work and, in particular, the importance of whose research in the topics treated in this book are widely acknowledged.

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February 22, 2010

The Poincaré inequality: W^{1,p} vs. W_0^{1,p}

Filed under: Giải Tích 6 (MA5205), Giải tích 8 (MA5206), Nghiên Cứu Khoa Học, PDEs — Tags: — Ngô Quốc Anh @ 1:50

In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, direct methods of the calculus of variations. A very closely related result is the Friedrichs’ inequality.

This topic will cover two versions of the Poincaré inequality, one is for W^{1,p}(\Omega) spaces and the other is for W_o^{1,p}(\Omega) spaces.

The classical Poincaré inequality for W^{1,p}(\Omega) spaces. Assume that 1\leq p \leq \infty and that \Omega is a bounded open subset of the n–dimensional Euclidean space \mathbb R^n with a Lipschitz boundary (i.e., \Omega is an open, bounded Lipschitz domain). Then there exists a constant C, depending only on \Omega and p, such that for every function u in the Sobolev space W^{1,p}(\Omega),

\displaystyle \| u - u_{\Omega} \|_{L^{p} (\Omega)} \leq C \| \nabla u \|_{L^{p} (\Omega)},

where

\displaystyle u_{\Omega} = \frac{1}{|\Omega|} \int_{\Omega} u(y) \, \mathrm{d} y

is the average value of u over \Omega, with |\Omega| standing for the Lebesgue measure of the domain \Omega.

Proof. We argue by contradiction. Were the stated estimate false, there would exist for each integer k = 1,... a function u_k \in W^{1,p}(\Omega) satisfying

\displaystyle \| u_k - (u_k)_{\Omega} \|_{L^{p} (\Omega)} \geq k \| \nabla u_k \|_{L^{p} (\Omega)}.

We renormalize by defining

\displaystyle {v_k} = \frac{{{u_k} - {{({u_k})}_\Omega }}}{{{{\left\| {{u_k} - {{({u_k})}_\Omega }} \right\|}_{{L^p}(\Omega )}}}}, \quad k \geqslant 1.

Then

\displaystyle {({v_k})_\Omega } = 0, \quad {\left\| {{v_k}} \right\|_{{L^p}(\Omega )}} = 1

and therefore

\displaystyle\| \nabla v_k \|_{L^{p} (\Omega)} \leqslant \frac{1}{k}.

In particular the functions \{v_k\}_{k\geq 1} are bounded in W^{1,p}(\Omega).

By mean of the Rellich-Kondrachov Theorem, there exists a subsequence {\{ {v_{{k_j}}}\} _{j \geqslant 1}} \subset {\{ {v_k}\} _{k \geqslant 1}} and a function v \in L^p(\Omega) such that

\displaystyle v_{k_j} \to v in L^p(\Omega).

Passing to a limit, one easily gets

\displaystyle v_\Omega = 0, \quad {\left\| {{v}} \right\|_{{L^p}(\Omega )}} = 1.

On the other hand, for each i=\overline{1,n} and \varphi \in C_0^\infty(\Omega),

\displaystyle\int_\Omega {v{\varphi _{{x_i}}}dx} = \mathop {\lim }\limits_{{k_j} \to \infty } \int_\Omega {{v_{{k_j}}}{\varphi _{{x_i}}}dx} = - \mathop {\lim }\limits_{{k_j} \to \infty } \int_\Omega {{v_{{k_j},{x_i}}}\varphi dx} = 0.

Consequently, v\in W^{1,p}(\Omega) with \nabla v=0 a.e. Thus v is constant since \Omega is connected. Since v_\Omega=0 then v \equiv 0. This contradicts to \|v\|_{L^p(\Omega)}=1.

The Poincaré inequality for W_0^{1,2}(\Omega) spaces. Assume that \Omega is a bounded open subset of the n-dimensional Euclidean space \mathbb R^n with a Lipschitz boundary (i.e., \Omega is an open, bounded Lipschitz domain). Then there exists a constant C, depending only on \Omega such that for every function u in the Sobolev space W_0^{1,2}(\Omega),

\displaystyle \| u \|_{L^2(\Omega)} \leq C \| \nabla u \|_{L^2(\Omega)}.

Proof. Assume \Omega can be enclosed in a cube

\displaystyle Q=\{ x \in \mathbb R^n: |x_i| \leqslant a, 1\leqslant i \leqslant n\}.

Then for any x \in Q, we have

\displaystyle\begin{gathered} {u^2}(x) = {\left( {\int_{ - a}^{{x_1}} {{u_{{x_1}}}(t,{x_2},...,{x_n})dt} } \right)^2} \hfill \\ \qquad\leqslant ({x_1} + a)\int_{ - a}^{{x_1}} {{{({u_{{x_1}}})}^2}dt} \hfill \\ \qquad\leqslant 2a\int_{ - a}^a {{{({u_{{x_1}}})}^2}dt} . \hfill \\ \end{gathered}.

Thus

\displaystyle\int_{ - a}^a {{u^2}(x)dx} \leqslant 4{a^2}\int_{ - a}^a {{{({u_{{x_1}}})}^2}dt}.

Integration over x_2,...,x_n from -a to a gives the result.

The Poincaré inequality for W_0^{1,p}(\Omega) spaces. Assume that 1\leq p<n and that \Omega is a bounded open subset of the n-dimensional Euclidean space \mathbb R^n with a Lipschitz boundary (i.e., \Omega is an open, bounded Lipschitz domain). Then there exists a constant C, depending only on \Omega and p, such that for every function u in the Sobolev space W_0^{1,p}(\Omega),

\displaystyle \| u \|_{L^{p^\star} (\Omega)} \leq C \| \nabla u \|_{L^{p} (\Omega)},

where p^\star is defined to be \frac{np}{n-p}.

Proof. The proof of this version is exactly the same to the proof of W^{1,p}(\Omega) case.

Remark. The point u =0 on the boundary of \Omega is important. Otherwise, the constant function will not satisfy the Poincaré inequality. In order to avoid this restriction, a weight has been added like the classical Poincaré inequality for W^{1,p}(\Omega) case. Sometimes, the Poincaré inequality for W_0^{1,p}(\Omega) spaces is called the Sobolev inequality.

October 28, 2009

The weak and weak* topologies: A few words

Filed under: Giải tích 8 (MA5206), Linh Tinh, Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 2:48

The weak and weak* topologies are the weakest in which certain linear functionals are continuous.

We start with a normed linear space X. The dual space of X, denoted by X', is the collection of all continuous linear functionals, i.e., the set of all mapping \ell : X \to \mathbb R satisfying

\ell(ax)=a \ell (x), \ell(x+y)=\ell(x)+\ell(y)

and

\displaystyle\lim_{n \to \infty} \ell(x_n) = \ell(x) when \displaystyle\lim_{n \to \infty} \|x_n - x\|=0.

Definition 1. In X, the strong topology is the norm topology, i.e., we can talk about an open set of X, for example U in the following sense: U \subset X is said to be open if and only if for each x_0 \in U, there exists \varepsilon>0 such that \{ x \in X: \|x-x_0\|<\varepsilon\} \subset U.

Claim 1. Bounded linear functionals are continuous in the strong topology.

Proof. We first recall that a linear functional \ell is said to be bounded if there is a positive number c such that |\ell (x)| \leq c\|x\| for all x \in X.

Now we assume \ell is continuous but not bounded; then for any choice of c=n, one has \ell(x_n) > n \|x_n\|. Clearly, x_n can be replaced by any multiple of x_n; if we normalize x_n so that

\displaystyle \|x_n\|=\frac{1}{\sqrt{n}}

then x_n \to 0 but \ell (x_n) \to \infty. This shows the lack of boundedness implies the lack of continuity.

Now we assume \ell is bounded. For arbitray x_n and x, one gets

|\ell(x_n)-\ell (x)| = |\ell (x_n-x)| \leq c\|x_n-x\|;

this shows that boundedness implies continuity.

Definition 2. In X, the weak topology is the weakest topology in which all bounded linear functionals are continuous.

The open sets in the weak topology are unions of finite intersections of sets of the form

\{ x : a< \ell(x) < b\}.

Clearly, in an infinite-dimensional space the intersection of a finite number of sets of the above form is unbounded. This shows that every set that is open in the weak topology is unbounded. In particular, the balls

\{ x : \|x\|<R\}

opens in the strong topology, are not open is the weak topology.

Definition 3. In X' the dual space of X, the weak* topology is the crudest topology in which all linear functionals

x: X' \to \mathbb R, x(\ell) := \ell(x)

are continuous.

If X' is nonreflexive, the weak* topology is genuinely coarser than the weak topology, as will be clear from the following theorem due to Alaoglu

Theorem (Alaoglu). The closed unit ball in X' is compact in the weak* topology.

We end this topic by the following theorem

Theorem. The closed unit ball in X is compact in the weak topology if and only if X is reflexive.

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