Questions tagged [weak-topology]

Let $X=(X,\tau)$ be a topological vector space whose continuous dual $X^*$ separates points (i.e., is T2). The weak topology $\tau_w$ on $X$ is defined to be the coarsest/weakest topology (that is, the topology with the fewest open sets) under which each element of $X^*$ remains continuous on $X$.

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21 views

Norm continuity of weakly continuous bounded holomorphic function

Let $X$ be a Banach space and consider a function $f\colon\overline{S}\to X$ where $S$ is the strip $$S=\{z\in\mathbb{C}:0<\operatorname{Re}z<1\}.$$ Suppose that $f$ is continuous with respect ...
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32 views

Unable to find weak topology T in $\mathbb{R}$

I am self studying topology and I was unable to solve this question. Wayne Patty's Question 2.4.5. Let $U$ be usual topology on $\mathbb{R}$ and let $T$ be the weak Topology on $\mathbb{R}$ induced ...
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23 views

How to describe weak topologies T on the $\mathbb{R}$

I was unable to solve this question of Exercise 2.4 (Weak Topologies) Question : Let U be the usual topology on $\mathbb{R}$ . Describe the weak topology T on $\mathbb{R}$ induced by each of the ...
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58 views

Compactness of a subset of $\ell^2$

Let $K \subset \ell^2(\mathbb{N})$ be a set defined as follows: $$ K := \left\{x = (x_1, x_2, \dots) \in \ell^2(\mathbb{N}) \,|\, |x_n| \le \frac{1}{n}\right\}.$$ Since $\ell^2(\mathbb{N})$ is a ...
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103 views

Is a relatively weakly compact subset necessarily norm bounded (by the Banach-Steinhaus theorem)?

Let $E_i$ be a normed vector space (EDIT: Assume $E_2$ is complete), $\langle\;\cdot\;,\;\cdot\;\rangle$ be a duality pairing between $E_1$ and $E_2$ with $$\left|\langle x_1,x_2\rangle\right|\le\left\...
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15 views

When continuity implies weak continuity

Let $X$ be a separable Hilbert space. Let $f:X\rightarrow \mathbb{R}$ be (lower semi-) continuous with respect to the weak topology. Then, I have read that if in addition, $f$ is convex, then it is ...
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19 views

The difference between 'weak limit point' and 'converge weakly'

For the following theorem. Let $S$ be a nonempty subset of $H$ and let $x:[0,+ \infty) \rightarrow H$. Assume that $\quad$ (i) for every $z\in S$, $\lim_{t\rightarrow \infty} \left\|x(t)-z\right\|$ ...
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41 views

prove weak topology is locally convex

Prove Banach space X endowed with weak topology is locally convex. To prove it has local convex base is easy since we can write down the neighborhood explicitly To question is do we need to prove the ...
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32 views

Conditions for continuity of the map $x \mapsto \mu_x$ (where $ \mu = \intop \mu_x d \mu (x)$ is the ergodic decomposition of $\mu$)

Let $(X,\mathcal{B},\mu,T)$ be a measure preserving system on a Borel space. Let $\{\mu_x\}_x$ be the conditional measures from the ergodic decomposition of $\mu$ (that is, $ \mu = \intop \mu_x d \mu ...
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120 views

Show that the characteristic function of a finite signed measure on a normed vector space is uniformly continuous

Let $E$ be a normed $\mathbb R$-vector space, $\mu$ be a finite signed measure on $(E,\mathcal B(E))$ and $$\hat\mu:E'\to\mathbb C\;,\;\;\;\varphi\mapsto\int\mu({\rm d}x)e^{{\rm i}\varphi}$$ denote ...
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compactness of probability measures in the weak topology for a non-metrizable compact space

I saw and understood some proofs which showed, that for a compact metric space $X$ the space of probability measures $\mathcal{P}(X)$ with the weak topology ($\mu\mapsto\int f\,\mathrm{d}\mu$ is ...
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50 views

Questions related to total set of linear functionals

Following questions are from Linear Operators edited by Nelson Dunford and Jacob T. Schwartz, Chapter V.7, Problem 5 and Problem 7. Let $X$ be a topological vector space (call its topology $\tau$). ...
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48 views

If $\|f_n\| \leq c$ and $f_n \rightarrow f$ in weak-star topology, then $\|f\| \leq c$?

Let $E=(E,\|\cdot \|_E)$ be a Banach space over $\mathbb{C}$. Consider $E'$ the dual of $E$, with norm $\|\cdot\|_{E'}$. Let $(f_n)_{n \in \mathbb{N}} \subset E'$ such that $f_n \rightarrow f$ in the ...
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46 views

An application of the Baire category theorem

In the highlighted sentence, $ K_n$ is a closed subset of $V^o$, which is the polar of $V$ and so is compact in the weak-* topology by the Banach-Alaoglu theorem. Therefore, $K_n$ is weak-* compact. ...
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44 views

Metrizability of the set of probability measures with the Kantorovich–Rubinstein metric

Let $X$ be a compact metric space and let $\mathcal{M}(X)$ be the set of probability measures. Then the weak topology on $\mathcal{M}(X)$ is metrizable, for example with the Wasserstein-metric and ...
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21 views

Continuity of integrals in the product of weak topologies

Consider two compact metrizable spaces $X$ and $Y$ and the sets of probability measures $\mathcal{M}(X)$ and $\mathcal{M}(Y)$ equipped with the weak topology. Furthermore let $f:X\times Y\to\mathbb{R}$...
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29 views

Compact set in the weak topology

Let $A \subset E$ be a subset of a Banach space $E$. I wonder if it is true that if $A$ is a compact set with the weak topology $\sigma(E, E')$, then $A$ is bounded.
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33 views

weakly continuous vs weakly sequentially continuous operator

Im reading some papers where the condition of weak sequential continuity is crucial, and this question come to my mind: Why it's more interesting to use this condition instead of the classical weak ...
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23 views

Usage of “weakly” when dealing with properties in the weak topology

Sometimes, when I read about topologies, for example the weak topology, they usually say "converges weakly", or "weakly continuous", "weakly open", etc., something ...
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38 views

Relationship between a Banach Space $X$ and its bidual $X^{**}$

Suppose $X$ is a Banach Space and $X^{**}$ is the bidual, $J:X\to X^{**}$ is the canonical map from $X$ to $X^{**}$. It's well known that $J$ might not be surjective. My question is that: given $\phi \...
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1answer
118 views

Weak topology is not metrizable

I'm now reading some properties of weak topology, I have some problems which may related to the topology property in non-metrizable space ($E $ is a Banach Space): I know that $E^*$ with weak* ...
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1answer
35 views

Weak topology Banach space with separable dual

Let $B$ be a Banach space with separable dual and let $(f_n)$ be dense and countable in $B^*$. Let $\tilde{\tau}$ be the initial topology associated to the collection of maps $f_n : B\rightarrow \...
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68 views

Weak topology of normed space

Let $X,Y$ be two normed space and $T:X\rightarrow Y$ be bounded linear operator.Now consider $X,Y$ with weak topology. My question is that does $T$ maps weakly compact set of $X$ to weakly compact set ...
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16 views

Continuity of extension of a continuous function

Let $X,Y$ be compact metrizable space and endow the space $\Delta(Y)$ of Borel probability measures $\nu: \mathscr{B}(Y) \to \mathbf{R}$ with the weak$^\star$ topology. Fix also a jointly continuous ...
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1answer
41 views

Is $L^P(\mu)$ norm continuous with respect to the underlying measure $\mu$?

Is $L^P(\mathbb{R}^d,\mu)$-norm continuous with respect to the underlying measure $\mu$ in the weak topology? Formally, let $\mu_n$ be a sequence of compactly supported Borel probability measures, ...
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42 views

Relative weak-star topology Vs product topologies

Let $S$ be a topological space. Denote by $\Delta(S)$ the set of probability measures on the Borel $\sigma$-algebras of $S$, endowed with its weak$^\star$-topology $\tau_S$. Question. Let $X,Y$ be ...
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1answer
14 views

Weak compactness of nonnegative part of unit ball of $L^1$

Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and let $B$ be the unit ball of $L^1(\Omega,\mathfrak{A},\mu;\mathbb{R})$. Suppose that $$ B_+=\big\{u\in B\;|\;u\geq 0\;\:\text{$\mu$-a.e.}\big\} $$ ...
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50 views

Weak Topology and the induced topology

Given a normed space $E$ with a subspace $M$, it is known that the weak topology on $M$ is the same as the induced topology of the weak topology on $E$. Why is this the case? From the Hahn-Banach ...
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1answer
48 views

Proof of Banach Alaoglu theorem by Brezis

I am trying to understand the proof of the Banach Alaoglu theorem in the functional analysis book by Brezis. These are the steps I don't quite follow. Let $Y = \mathbb{R}^E$, a set of functions from $...
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1answer
30 views

You can prove that compacts in the weak topology are limited without the Eberlein-Šmulian theorem

You can prove that compacts in the weak topology are bounded without the Eberlein-Šmulian theorem? With Smuliam's Theorem this is immediate, because if you assume by contradiction that $K$ is unborded ...
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1answer
60 views

$\text{dim}\;E<\infty$ if and only if $E'$, in the weak topology, is normable

Let $E$ be a locally convex Hausdorff space over $\mathbb{C}$. I want to prove that: $\text{dim} \;E<\infty$ if and only $E'_{\sigma}$ is normable, where $E'_{\sigma}$ denotes the dual space of $E$ ...
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1answer
64 views

Continuity in weak topology implies continuity in strong topology

What I want to prove is the following: Let $X$ and $Y$ be normed vector spaces. If a linear operator $T:(X, \sigma(X,X^*))\to (Y, \sigma(Y,Y^*))$ is continuous, then $T:(X,\|\cdot\|)\to(Y,\|\cdot\|)$ ...
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1answer
88 views

Why is the weak topology not more widely defined?

I first came across the concept of a weak topology (or initial topology) in the context of functional analysis. It is virtually unmentioned in such standard upper level undergraduate / first year ...
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42 views

Looking for a function that is continuous but not sequentially weakly continuous

Let $(X, \|\cdot\|) $ be a Banach space. A function $g:X \longrightarrow X$ is said to be sequentially weakly continuous if for every sequence $(x_n)$ in $X$ such that $x_n \rightharpoonup x$, we have ...
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45 views

K(subset of C[0,1]) does not attain the minimal norm [closed]

Let $V=C[0,1]$ and $K\subset V$ be defined by $K= \{f \in V | \int_{0}^{1/2} f(t) dt - \int_{1/2}^{1} f(t) dt = 1\}$. Show that $K$ does not admit an element with minimal norm.
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35 views

$l_1$ has no infinite dimensional subspace that is reflexive. [closed]

How to show that $l_1$ does not contain an infinite dimensional subspace that is reflexive.
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21 views

Weak* continuous functions [duplicate]

I want to prove that in a normed space $E$, every weak* continuous, linear operator $T:E^*\rightarrow\mathbb{R}$ is in the form $T_x$ (where $T_x$ is the operator "evaluation in $x$", that ...
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19 views

Does quasiconvexity imply weak lower semicontinuity?

Let $W^{1,p}(\Omega)$ be the usual Sobolev space, $f(x,\eta,\xi)$ a continuous function, $1<p<\infty$, $\Omega$ an open and bounded subset of $\mathbb R^n$, and $F: W^{1,p}(\Omega) \...
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1answer
31 views

Question in a theorem related to topic weak topology

I have done a course on topology and I am self studying concepts that were not taught in class from the book Foundations of Topology ( C Wayne Patty). I got struck upon this theorem. My ...
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15 views

Why $C\mapsto s(x^*,C)$ is affine function?

Let $X$ be a separable Banach space, the associated dual space is denoted by $X^*$ and the usual duality between $X$ and $X^*$ by $\langle , \rangle$. For $C$ nonempty weakly compact convex subsets of ...
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11 views

$\overline{\text{co}}(\cup_nC_n)$ is weakly locally compact?

Let $X$ be a separable Banach space such that $X$ and its dual $X^*$ have Radon-Nikodym property. Let $\{C_n\}$ be a family of convex, weakly compact subsets and bounded subset of $X$. Take $A=\...
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35 views

$C$ is **weakly compact** or **weakly locally compact**?

Let $X$ be a separable Banach space such that $X$ and its dual $X^*$ have Radon-Nikodym property. Let $C$ be a convex, closed and bounded subset of $X$. Can we say that $C$ is weakly compact or ...
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1answer
26 views

$ \sup_{x\in C}\|x\|=\sup_{x^*\in B^*}\sup_{x\in C}\langle x^*,x\rangle $

Let $X$ be a separable Banach space, the associated dual space is denoted by $X^*$ and the usual duality between $X$ and $X^*$ by $\langle , \rangle$. Let $B^*$ the closed unit ball of $X^*$. Take $...
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1answer
18 views

$\{s(. , C_n)\}$ is equicontinuous on $X^*$.

Let $X$ be a separable Banach space, the associated dual space is denoted by $X^*$ and the usual duality between $X$ and $X^*$ by $\langle , \rangle$. For $C$ nonempty weakly compact convex subsets of ...
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1answer
53 views

$ \frac{1}{n}\sum_{i=1}^{n}{C_i}\text{ is weakly compact} $

Let $X$ be a separable Banach space. Let $C_1,...,C_n$ are nonempty weakly compact convex subsets of $X$. Why $$ \frac{1}{n}\sum_{i=1}^{n}{C_i}\text{ is weakly compact} $$ An idea please.
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10 views

$\cap_p\overline{co}\cup_{m\geq p}{ \frac{1}{m}\sum_{n=2}^mC_n}\subset \cap_p\overline{co}\cup_{m\geq p}({\overline{co}\cup_{n=2}^mC_n})$

Let $X$ be a separable Banach space and by $ w$ we shall indicate the weak topology on $X$. Let $\mathcal{P}_{wkc}(X)$ be the collection of all nonempty $w$-compact convex subsets of $X$. Let $\{C_n\...
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32 views

What exactly is the relationship between weak* convergence and sequential convergence?

I am studying Banach algebras, and the theory frequently makes use of the weak* topology which I am not very familiar with. As I understand it, the weak* topology is defined as follows: let $X$ a ...
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15 views

Weak-${\star}$ Topology of Inductive Limit of Closed Unit Balls?

This is purely out of curiosity, but I ask. Banach Alaoglou the closed unit ball $\overline{B_1}$ in the weak$^{\star}$ topology of any Banach space $X$ is compact. One can, therefore, construct ...
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17 views

Weak* cluster points

Let $X$ be a banach space. I know that a convergent sequence $\{f_n\}\subset X^*$ in the weak* topology on $X^*$ is a sequence such that for all $x\in X$, $\{f_n(x)\}_{n=1}^{\infty}$ is convergent in $...
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22 views

How can I conclude that the weak topology on $~\mathcal l^2~$ is a proper subset of the norm topology from what I've done?

I'm doing a problem in topology. In a) I proved that the weak topology is coarser than the norm topology, and in b) I proved that the standard one sequence $~(e_n)~$ in $~\mathcal l^2~$ approaches $~(...

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