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Questions tagged [weak-topology]

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3answers
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How to prove or disprove the following statement on Hausdorff topology

Let $X$ be a set and let $Y$ be a Hausdorff space. Let $f\colon X \to Y$ be a given mapping. Define $U \subset X $ to be open in $X$ if, and only if $U = f^{-1}(V)$ for some set $V$ open in $Y$. This ...
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2answers
37 views

Weak continuity of the addition and scalar multiplication

Let $X$ be an infinite dimensional normed vector space. Show that vector addition and scalar multiplication are weakly continuous. $$+:X×X \rightarrow X; +(x,y)=x+y$$ $$•:\mathbb {R}×X \rightarrow X; •...
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2answers
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How to prove that following topology is Hausdorff?

The weakest topology on $\mathbb{R}$ such that all polynomials (in a single variable) are continuous, is Hausdorff. How to think? I have no idea how to start with.
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1answer
42 views

Weak-* continuous functionals with bounded level sets

Let $X$ be an infinite-dimensional Banach space, $X^*$ its topological dual and $f:X^*\to\mathbb{R}$ some weak-* continuous functional (not necessarily linear). Is it possible for $f^{-1}(a)$ to ...
2
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1answer
16 views

Can you have a weakly convergent sequence of unbounded linear operators? (Example)

Is it at all feasible to have a weakly convergent sequence of unbounded linear operators? If so, what is a concrete example of a sequence of not necessarily bounded linear operators who converge in ...
2
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2answers
41 views

Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$

Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$ My trial I'm thought of it this way that if $g_n(x)=ne^{-nx}$ converges weakly in $L_1(\Omega)$ then it is bounded....
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2answers
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$+:X\times X\to X,(x,y)\mapsto +(x,y)=x+y$ and $\cdot:\Bbb{R}\times X\to X,(\lambda,x)\mapsto \cdot(\lambda,y)=\lambda\cdot x$ are weakly continuous

$$+:X\times X\to X,\\(x,y)\mapsto +(x,y)=x+y$$ and $$\cdot:\Bbb{R}\times X\to X,\\(x,y)\mapsto \cdot(\lambda,y)=\lambda\cdot x$$ are weakly continuous, where $X$ is an infinite dimensional normed ...
3
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1answer
38 views

A question on Weak and Norm topology

How to prove that If $F$ is closed with respect to the weak topology then $F$ is closed with respect to the norm topology... Weak topology means let $V$ Banach space the weak topology on $V*$ is ...
2
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0answers
21 views

Weak-continuity of an operator on the vector space of measures

Hello and thanks in advance for your time. Let $\mathcal{S} \subset \mathbb{R}^d$ (we can also choose $\mathcal{S}$ compact if this helps here) for some integer $d$ and let $M(\mathcal{S})$ be the ...
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2answers
37 views

Unit ball of $X^{**}$ is weakly compact!

Is it true that the closed unit ball in $X^{**}$ is compact with respect to the weak topology on $X^{**}$, where $X$ is a Banach space? If so, how can we prove it?
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2answers
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If $Y\subset X$ are Banach spaces such that $Y$ is dense in $X$, is it true that $X'$ is dense in $Y'$?

If $Y$ is a dense subspace of a Banach space $(X,\|\cdot\|_1)$ and $(Y,\|\cdot\|_2)$ is a Banach space such that the inclusion from $(Y,\|\cdot\|_2)$ into $(X,\|\cdot\|_1)$ is continuous, then it is ...
1
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1answer
24 views

Rudin's functional analysis theorem 3.21.

Small proof of Suppose $X$ is a topological vector space on which $X^*$ separates points. Suppose $A$ and $B$ are disjoint, nonempty, compact, convex sets in $X$. Then there exits $\Lambda \in X^*$ ...
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2answers
26 views

Literature on relationship between strong and weak* (weak star) convergence

I am trying to follow a proof in the paper Wasserstein Generative Adversarial Networks by Arjovsky et al. (proof A in supplementary material). They show that the convergenc of the total variation ...
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0answers
17 views

example of a Banach space X and a subspace Y such that Y is strongly closed but not wealky closed.

I am trying to solve this exercise: find a Banach space X and a subspace Y such that Y is strongly closed but is not wealky closed. I know Y can't be a convex subspace because strongly closed + ...
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2answers
44 views

A weakly convergent sequence in a compact set, is strongly convegnet

Let $E$ be a Banach space, and $K \subset E$, compact set for the strong topology. And let $(x_n)_n$ converges for the weak topology $\sigma(E,E^*)$ to $x$. Why $(x_n)_n$ converges for the strong ...
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1answer
39 views

Compactness in weak$^*$-topology on $l_1(\mathbb{N})$

Let $K$ denote the closed unit ball of $l_1(\mathbb{N})$ (considered as a vector space over $\mathbb{C}$). Is $\mathrm{co}(\mathrm{Ext}(K))$ (the convex hull of its extreme points) compact in the ...
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0answers
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Relation between energy functionals of stationary and dynamic version of a PDE.

Actually, I am dealing with a very specific equation, but It is rather complex and goes along with many notations and long calculations so I would like to ask a question in a conceptual spirit. Given ...
1
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1answer
32 views

Proof theorem 3.17 Rudin's functional analysis

Consider the two theorems (3.15 and 3.16) of Rudin's functional analysis: Theorem 3.15: If $V$ is a neighborhood of $0$ in a topological vector space $X$ and if $$K = \left\{\Lambda \in X^* : |\...
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2answers
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Is a metrizable and compact subset of $E^*$ is a metrizable complete subset ? for the weak* topology

If we have a Banach space $E$. and we consider it's dual $E^*$, it is a Banach space. so we consider the weak* topology ($\sigma(E^*,E)$) on $E^*$. So My question is : If we have a set $B\subset E^*$,...
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0answers
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Closed unit ball and convex hull of its extreme points

Let $K$ denote the closed unit ball of $l_1(\mathbb{N})$ (considered as a vector space over $\mathbb{C}$). (a) Show that the set of extreme points of $K$ is $\mathrm{Ext}(K)= \{ \lambda e_n :\lambda ...
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1answer
25 views

The image of the weak topology with the canonical injection $J$

Let $E$ be a Banach space. With the weak topology $\sigma(E,E^*)$. And let $J:E\rightarrow E^{**} $ be the canonical injection. Can we prove that the image of the weak topology with $J$ is exactly ...
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2answers
37 views

Rudin's functional analysis theorem 3.12

Suppose $E$ is a convex subset of a locally convex space $X$. Then the weak closure $\overline{E}_w$ of $E$ is equal to its original closure $\overline{E}$. The proof starts as follows $\overline{...
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0answers
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Rudin's functional analysis theorem 3.10, proof that multiplication is continuous

Suppose $X$ is a vector space and $X'$ is a separating vector space of linear functionals on $X$. Then the $X'$-topology $\tau'$ makes $X$ into a locally convex space whose dual space is $X'$. ...
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2answers
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Why are weak topologies useful in functional analysis?

I've been reading through chapter 3, Rudin's Functional Analysis, and an important point is the one of weak topology. From the theorems it seems to me weak topologies are somehow the result of ...
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0answers
38 views

A set $C$ that is not closed in the weak* topology

I have a problem, but first I would like to understand your statement. The statement is about a subset $C\subset l^{\infty}$ that is convex and closed in $C(l^{\infty}, |.|_{sup})$. I know that if $C$ ...
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2answers
39 views

Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology.

Let X be a Banach space. Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology. Could anybody give me some hints to start?
2
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1answer
42 views

Rudin functional analysis, section 3.8 (Topological preliminaries)

A quote from section 3.8. of Rudin's functional analysis: Suppose next that $X$ is a set and $\mathcal{F}$ is a nonempty family of mappings $f: X \to Y_f$, where each $Y_f$ is a topological space (...
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0answers
52 views

Weak closure of subsets of the unitary sphere of a Banach space.

Assume that $(X,\|\cdot\|)$ is a Banach space with $\|\cdot\|$ strictly convex. Define $S=\{x\in X:\|x\|=1\}$. Suppose that $\varepsilon>0$ and $x_0\in S$ and define $$ B_\varepsilon=\{x\in X:\|x-...
3
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3answers
81 views

$L^1([0,1])$ closed unit ball is not weakly compact

So I would like to prove this result by constructing a sequence of functions $u_n$ in $L^1([0,1])$, such that $\|u_n\|_{L^1}\leq 1$ for all $n$, but this subsequence does not have a convergent ...
2
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1answer
30 views

$X^*$ with $weak^*$-topology is first category in itself when $X$ is infinite dimensional Fréchet space

I am reading Rudin's Functional Analysis and stuck at the problem in exercise 11 on page 87 : Let $X$ be infinite dimensional Fréchet space , then $X^*$ with it's $weak^*$-topology is of first ...
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0answers
25 views

Neighborhood of zero on weak topology

I have a homework, I have tried so much on this question, but I couldn't solve it. How can I prove it? Can you help me? Thank you. $E$ is a normed space and $E'$ is the topological dual of $E$. If $...
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1answer
21 views

For an infinite dimensional Banach space, $X^*$ when given the weak* topology is of the first category in itself [duplicate]

Let $X$ be an infinite dimensional Banach space. Why is $X^*$ of the first category in itself when given the weak* topology. Very closely related to $X^*$ with its weak*-topology is of the first ...
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2answers
95 views

Question about a weak*-norm continuity of a linear operator

Let $X$ and $Y$ be infinite dimensional normed linear spaces and let $S:Y^*\to X^*$ be a one-one linear operator. I want to show that $S$ can not be weak*-norm continuous. My idea is to choose a ...
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1answer
52 views

Closed convex hull of pure states of non-unital $C^*$-algebras

It is known that, when $\mathcal{A}$ is a $C^{*}$-algebra with an identity element, the space $\mathcal{S}$ of states of $\mathcal{A}$ is a convex subset of the topological dual $\mathcal{A}^*$ of $\...
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2answers
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Can we prove that in a convex space, Weakly closed=> weak* closed ??

We know that weak* Topology is smaller than weak topology. So weak* closed sets are weakly closed. Banach Mazur theorem says "Strongly closed implies weakly closed if space is convex." can we expect ...
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0answers
28 views

Is the space of finite Radon measures, equipped with weak topology, locally compact and second countable?

I saw someone in the following link claims the space of finite Radon measures is locally compact with weak topology. Is the space of finite Radon measures, equipped with weak topology, locally ...
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1answer
24 views

If for every $e\in E\setminus\{0\},$ there exists $e^*\in A$ such that $e^*(e)\neq 0,$ then $A$ is weak-star dense in $E^*?$

Suppose that $E$ is a Banach space and $E^*$ is a dual space of $E,$ that is, $E^*$ contains all bounded linear functionals on $E.$ Let $A$ be a closed subspace of $E^*.$ Equip $E^*$ with weak$^*$-...
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1answer
53 views

Space of measures has the Dunford-Pettis property?

I'm trying to prove Corollary 5.4.6. from Albiac, Kalton - Topics in Banach Space Theory: If $ K $ is a compact Hausdorff space then $ \mathcal{M}(K) $ has (DPP). I want to use The Dunford-...
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1answer
102 views

Compactness / sequentially compact / reflexive space / separable spaces / weak topology

I'm a bit confuse with all theses notions. Let $E$ a normed vector space of infinite dimension (also Banach, but it's probably not important). The theorem of Eberlin Smulian theorem says that : all ...
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2answers
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In “Analyse fonctionnelle” of Brezis, in chapter III why do we need Banach spaces ? (especially for Kakutani's theorem)

In the book of Brezis : "Analyse fonctionnelle : Théorie et application", chapter III (i.e. construction of weak topology, weak-* topology reflexives spaces...), why do we need "Banach spaces" ? Isn't ...
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1answer
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When $E$ has finite dimension, then the weak topology and the strong topologie are the same.

Let $E$ a normed vector space of finite dimension. Then the strong topology and the weak topology are the same. To prove it, we take $x_0\in E$ and $U$ an open set (for the strong topology) that ...
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1answer
57 views

When does the continuous dual of the weak operator topology consist only of finite linear combinations of evaluations?

Let $V$ and $W$ be topological vector spaces, say, over $\mathbb C$. Let $L(V, W)$ be the vector space of continuous linear maps equipped with the weak operator topology, which is the initial topology ...
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1answer
32 views

Is $E^{\prime\prime}$ weak*-separable for separable Banach spaces?

My reasoning so far is as follows: Let $E$ be a separable Banach space. Since ${\mathrm{id}:(E,\|\cdot\|)\longrightarrow(E,w)}$ is a continuous surjection and $(E,\|\cdot\|)$ is separable, so too is $(...
2
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1answer
49 views

Examples of elements in $(X^*)^*$ that are not evaluation maps

I just learned about the weak-* topology on $X^*$, and in this context was introduced to the 'dual of the dual of a space', and the functional $J: X \rightarrow (X^*)^*$ where $J(x)[\psi] = \psi(x)$, ...
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2answers
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Linear functional $\psi:X \to \mathbb{R}$ is $W$-weakly continuous if and only if it belongs to $W$

I'm trying to understand the proof of the following proposition from Royden and Fitzpatrick (Chapter 14) - the step that I highlight is in which I would like confirmation on my justification. I ...
2
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2answers
101 views

weak topology has less open set than strong topology (in Banach spaces). Why?

Let $E$ a Banach spaces of infinite dimension. The weak topologie is the thickest that makes functional continuous. Let denote $\mathcal T_W$ the weak topology on $E$. 1) I call "Dual topological" ...
3
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1answer
49 views

Does the set of characters, $\Omega(\mathcal{A})$, over a C${}^{\ast}$-algebra, $\mathcal{A}$, generate a weakly dense subspace of $\mathcal{A}'$?

Let $\mathcal{A}$ be an abelian C${}^{\ast}$-Algebra with unit. We know that $\mathcal{A}\cong C(\Omega(\mathcal{A}))$, where $\Omega(\mathcal{A})\subseteq\mathcal{A}'_{\geq 0}$. Note that for $\...
3
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1answer
76 views

Do operators from $L(X', Y')$ preserve weak*-convergence?

I am wondering whether the following is true: Let $X, Y$ be normed spaces and $T \in L(X', Y')$. If $x_n' \overset{*}{\rightharpoonup} x'$ in $X'$, then $Tx_n' \overset{*}{\rightharpoonup} Tx'$ ...
2
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2answers
50 views

Does the notion “weak convergence” coincide with that using in functional analysis?

Let $\mu_k$ be a sequence of probability (Borel) measures on $\mathbb{R}^n$. We say, $\mu_k$ converges to a probability measure $\mu$ weakly if $\int gd\mu_k \to \int gd\mu$ for every continuous ...
0
votes
1answer
52 views

How to show a normed space $X$ is reflexive if the weak topology and weak* topology on $X^*$ are equal?

Let $X$ be a normed space such that the weak topology and weak* topology on $X^*$ are the same. I want to show $X$ is reflexive. My attempt is: Since we can use $\sigma(X^*,\widehat{X})$ to denote ...