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Questions tagged [weak-topology]

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1answer
33 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
17 views

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 ...
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1answer
23 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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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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67 views

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
23 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
27 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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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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36 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
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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?
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1answer
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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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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-...
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3answers
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$L^1([0,1])$ 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 ...
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1answer
29 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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23 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
20 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
77 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
45 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
36 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 ...
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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 ...
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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
43 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
66 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
193 views

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
35 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 ...
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1answer
53 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
31 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 $(...
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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
43 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 ...
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2answers
85 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" ...
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1answer
48 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 $\...
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1answer
67 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 $Tx_n' \overset{*}{\rightharpoonup} Tx'$ ...
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2answers
47 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 ...
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1answer
41 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
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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{...
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1answer
27 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 ...
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2answers
60 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, ...
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1answer
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[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 ...
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0answers
29 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(...
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1answer
43 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 ...
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0answers
50 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 ...
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1answer
66 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
82 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
70 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 $\...
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1answer
53 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 ...
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1answer
111 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 ...
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0answers
58 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 ...
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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 $...
1
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1answer
33 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 ...