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Questions tagged [weak-topology]

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2
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1answer
22 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
20 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
17 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 ...
3
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2answers
72 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 ...
1
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1answer
38 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
28 views

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 ...
0
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0answers
27 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 ...
0
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1answer
21 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
35 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-...
2
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1answer
47 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 ...
2
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1answer
34 views

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 ...
5
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1answer
52 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 ...
1
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1answer
30 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 $J(x)[\psi] = \psi(x)$, ...
0
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2answers
41 views

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
75 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
41 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
51 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
44 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
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1answer
34 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 ...
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1answer
33 views

Exercise on Sobolev Spaces and strong/weak convergence

I have to solve this exercise: "Fix $v \in \mathcal{C}^\infty_c(\mathbb{R})$. Discuss the strong and the weak convergence of the sequence $u_n(x) = \frac{v(nx-n^2)}{n}$ in the spaces $W^{k,p}(\mathbb{...
0
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1answer
26 views

weak topology generated by constant functions

Find the weak topology on R generated by the family of all constant functions. f : R → (R, Usual) I solve like below Subbasis of weak topology is {f^(-1)(U) s.t U is open in (R, Usual) let f(x)=a ...
2
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2answers
48 views

weak$^*$ topology is strictly coarser than weak topology ; a remark in Brezis

Let $E$ be a banach space. $\sigma(E^*,E)$ be the weak $^*$ topology, $\sigma(E^*, E^{**})$ the weak topology on $E^*$ induced by the functionals on $E^{**}$. It is proven in Brezis, pg65, ...
3
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1answer
64 views

[Looking for Confirmation](X,G) is a duality, then (X,G1) is also one iff G1 is dense in G

Edit: I noticed, that my original post has not had any replies, therefore I wrote the problem again (together with my attemps, which are also already more detailed that 2 days ago) and structured it ...
0
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0answers
25 views

Is the Unit ball of $X^{**} $ weak*- sequentially compact?

Am in the middle of a problem and i have the following conditions : Let $X$ be a reflexive Banach space with Schauder basis $(e_n)_{n=1}^{\infty}$, i have a sequence $x_n^{**} \in B_{X^{**}}\biggl(...
2
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1answer
37 views

Prove that weak operator topology in von Neuman Algebra implies norm topology in $C^*$ - algebra

I am a beginner in Operator Theory and Functional Analysis. On the space of bounded operators on Hilbert space $H$, We claim that " It is true that von Neumann algebras are $C^*$- algebras of ...
2
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0answers
45 views

Sequential Banach-Alaoglu theorem for a Bochner space

I have a question concerning the sequential weak-$*$ compactness in a Bochner space. Let $H$ be some non-separable Hilbert space. Consider the Bochner space $L^{\infty}(0,T;H)$. It is known that this ...
0
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1answer
58 views

Separable dual implies space is weakly metrisable

For $E$ a Banach space and $E'$ (the continuous dual) a separable space, show that the closed unit ball $\overline{B}_{E} \subset E$ is weakly metrisable. My work so far: As $E'$ is separable, we ...
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0answers
76 views

metrizability of weak topology

I have a problem with a line at the end of an example who tolds this: let $(E,||-||)$ be a separable Banach space. If there exist a weak Cauchy sequence who doesn't converge weakly to an element of $E$...
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0answers
64 views

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

The space of Radon measures on a complete separable metric space $E$, endowed with the Borel σ-algebra, is denoted by $\mathcal M(E)$, while $\mathcal M_F(E)$ is the subspace of finite measures in $\...
3
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1answer
43 views

Can a weakly Cauchy sequence in a non-complete inner product space be unbounded?

Let $V$ be a non-complete inner product space, and let $x_n$ be a weakly Cauchy sequence, i.e suppose $\langle x_n, y\rangle$ converges for every $y \in V$. Is it true $x_n$ is bounded? I know this ...
1
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1answer
106 views

How the norm-closed unit ball of $c_0$ is not weakly compact?

I saw an answer here: The norm-closed unit ball of $c_0$ is not weakly compact. But I am new to analysis. I couldn't understand this answer. All I know is someone is using net accumulation to get ...
4
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0answers
56 views

Soft question - a subset of a Hilbert space endowed with subspace topology

I am considering a Hilbert space $X$, endowed with its weak topology. I need to work with a subset (but not a subspace) $S$ of $X$. However I need to endow $S$ with the subspace topology (so $U$ is ...
1
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1answer
31 views

How to show $\hat{x}\in X^{**}$ without using $(X^*,wk^*)^*\subseteq X^{**}$?

Definition: Let $X$ be a topological vector space and let $x\in X$. Then $x$ defines a linear functional $\hat{x}$ on $X^*$ via $\hat{x}(f)=f(x)$ $(f\in X^*)$. From this definition we can easily get $...
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1answer
32 views

Finding a weak open neighbourhood of a Hilbert space with a certain property

Let $X$ a Hilbert space. I will write BWON for a basic weak open neighbourhood of $0$ in $X$. Let $U$ a BWON, and for each $1\leq i \leq n$, let $f_i: X \to X$ be a weakly continuous map. I want to ...
2
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2answers
126 views

Help understanding the weak topology on the dual of the Schwartz space?

I was working on the following problem, Prove that $\lim_{\epsilon \downarrow 0} \frac{x - x_0}{(x - x_0)^2 + \epsilon^2} = \mathcal{P}(1/(x - x_0)$, in the weak topology on $\mathcal{S}'(\mathbf{...
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0answers
31 views

Weak closed in Sobolev space

Let $\omega$ be a bounded open subset in $R^5$ and $b$ be in $L^2(\omega)$. Put $M = \{ u \in W^{1,3}(\omega): \int_\omega b.u^2= 1 \} $. Is $M$ weakly closed in $W_0^{1,3}(\omega)$ ?
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1answer
52 views

Mazur's theorem counterexample for the weak-$*$ topology

If $X$ is a banach space, the Mazur's theorem shows that the norm and the weak closure of a convex set coincide. The Mazur's theorem seems to be false for the weak-$*$ topology, that is, the norm ...
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0answers
49 views

If $X$ is finite dimensional, what is an explicit metric inducing the weak topology on $X?$

Theorem: Let $X$ be a normed linear space. Then weak topology on $X$ is metrizable if and only if $X$ is finite dimensional. I have proven the forward direction, which is, if weak topology on $X$ is ...
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0answers
28 views

Unit ball of the adjoint space of a separable Banach space is second-countable in the weak* topology.

In the proof : If $A$ is separable, then $\Delta(A)$ satisfies the second axiom of countability. ($\Delta(A)$ the set of all complex homomorphisms of $A$.) I have found they says "It is easy to see ...
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2answers
163 views

Unit ball in dual space is weak*separable

Let $X$ be a separable Banach space, $X'$ the dual space of $X$ endowed with the weak$*$-topology. How to prove that the unit ball $B$ in $X'$ is weak*-separable ? I only know Banach-Alaoglu, which ...
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0answers
48 views

When does two random measures coincide in distribution?

Given a Polish space $(E,d)$ with a sigma-algebra $\mathcal{E},$ we say that a family $\mathcal{A}\subset\mathcal{E}$ is a separating class if if two probability measures that agree on $\mathcal{A}$ ...
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16 views

Does $B_K = \overline{co}^{w^*}(ext(B_K))$ hold for any dual space $K?$

Notation: Given a space $X,$ we denote its closed unit ball by $B_X.$ Let $ext(X)$ be the set of extreme points of $X.$ Also, $\overline{co}(X)$ denote the closed convex hull of the set $X.$ Let $\...
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1answer
24 views

Element of weak closure by compact conditions

Let $G$ be a locally compact, metrizable and separable group, $X=L^\infty(G)$ and $A\subseteq X$ and $1\in X$. I know that forall $\varepsilon>0$ and all compact sets $C\subseteq G$ there exists an ...
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0answers
34 views

Question about a projection and weak topology

Let $X$ a Hilbert space, $U$ be a basic weak neighbourhood of $0$ in $X$, and P an orthogonal projection. I want to prove that there exists $\varepsilon >0$ such that for any $x\in X$ with $\|x\| \...
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0answers
41 views

A weakly open subset of the unit ball of the Read's space $R$ (an infinite-dimensional Banach space) is unbounded.

We know every weakly open subset of an infinite-dimensional Banach vector space X is unbounded. Now, Read's space $R$ (an infinite-dimensional Banach space) has the property: there is $ρ >0$ such ...
2
votes
1answer
72 views

Set of Positive Sequences that Sum to 1 is Compact under Product Topology?

I’m wondering if the set of non-negative sequences which sum to 1 is compact under the product (or weak) topologies. That is: $(a_1,a_2,...)$ such that $\sum_n a_n=1$ where $a_n \geq 0 \forall n$ I ...
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0answers
43 views

Implication of implying weak convergence: bounded ball w.r.t. some distance is contained in another bounded ball?

Consider Borel probability measures on $\mathbb R^d$. The the Prokhorov metric, $d_P$, metrizes weak convergence of probability measures, i.e., $d_P(Q_n,Q)\to 0$ if and only if $Q_n\to Q$ weakly. Let $...
0
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1answer
19 views

Suppose Ball $X^*$ (closed unit ball of $X^*$) is weak-star metrizable. To show $F^\perp\subseteq\bigcap\limits^{\infty}_{n=1}U_n$.

Let $X$ be a normed space. Suppose Ball $X^*$ (closed unit ball of $X^*$) is weak-star metrizable. Then there exists a metric $d$ on ball $X^*$ such that the weak-star topology on ball $X^*$ is ...