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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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Reflexitivity of sobolev spaces

Let $\Omega \subset \mathbb{R}^n$ open I'm interested to understand the reflexivity property of Sobolev spaces $W^{1, p}(\Omega)$ for $p \in [1, +\infty]$, using the fact that $L^p(\Omega)$ are ...
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Weak-star closed condition

In Dunford-Schwartz,Linear Operator, I, General Theory, to prove the Eberlein-Smulian theorem we use the following fact: Let $X$ be Banach space, Let $B$ be the norm closed unite ball of $X^*$ and $x^...
Manuel Bonanno's user avatar
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Operator continuous with range in weak topology iff continuous with range in the norm topology

I'm self studying MacCluer's book Elementary Functional Analysis, and I came across the following problem. Problem 3.18. Let $X$ and $Y$ be normed linear spaces and suppose $T \colon X \to Y$ is ...
Damalone's user avatar
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An interesting family of seminorms $\mathcal F$ and comparison between the topology generated by this seminorms and the Weak Operator Topology.

I am learning functional analysis and I am stuck with the following questions from Strong Operator Topology and Weak Operator topology on $\mathcal B(H)=\{T:H\to H:T$ is Op-Norm continuous,linear $\}$....
Kishalay Sarkar's user avatar
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Completeness of Y (normed tvs) in the proof of th. 2 in ch. V section 3 of Yosida's Functional Analysis ed 6

I'm stuck on the proof of theorem 2 in chapter V, section 3 of Yosida's Functional Analysis edition 6 (pages 140,141). Theorem 2 says : A locally convex linear topological space X is reflexive iff it ...
PTony's user avatar
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Exercise about convex hull and weak convegence

Suposse $X$ is a normed space, $(x_k)_{k \in \mathbb{N}}$ a sequence in $X$, $x \in X$ such that $x_k \underset{weakly}{\rightarrow} x$. Let $co$ denote the convex hull. Show that, there exists $y_k \...
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The base for the weak topology is actually a base

I am reading Real and Functional Analysis by Serge Lang and I am having problems with this: Let $Y$ be a topological space and let $\mathcal{F}$ be a family of mappings $f:X\rightarrow Y$ of $X$ into $...
Branco Flores Rocha's user avatar
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Weak closure of a subset of the unit sphere of $\ell_1$

It is a well-known and standard fact that for every infinite-dimensional Banach space $E$ the weak closure $\overline{S_E}^w$ of the unit sphere $S_E$ of $E$ is equal to the closed unit ball $B_E$ of $...
Damian Sobota's user avatar
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Understanding the definiton of weakly open sest and weak convergence

I am learning about weak and weak* topology. In the book I am reading the following is mentioned Definiton (weak topology) If X is a LCS, the weak topology on X, is the topoloty defined by the family ...
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Difference between the weak and weak* topology (using seminorms to define the topologies)

A few days ago, I was interested in the weak topology and the fact that the weak topology is the coarsest topology such that $f:X \rightarrow \mathbb{K}$ is continuous. (How to show that, the weak ...
Peter's user avatar
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How to show that, the weak topology is the coarsest topology such that all $f:E \rightarrow \mathbb{K}$ are continuous?

Let E be a normed space, $E':=\{f:E \rightarrow \mathbb{K}| f \text{ is continuous and linear}\}$. Define $p_f(x):=|f(x)| where f \in E'$ and $x \in E$. Consider the family of seminorms $\mathcal{P}=\{...
Peter's user avatar
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Is the strict topology stronger than the weak* topology on the Fourier-Stieltjes algebra?

Let $G$ be a locally compact group and $B(G)$ its Fourier-Stieltjes algebra. It can be defined as either the dual of the group C$^*$-algebra $C^*(G)$ or the linear span of continuous positive definite ...
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$B(X^{**})$ is the $w^*$-closure of $B(X)$ in $X^{**}$

I am reading Bollobás' Linear Analysis. Chapter 8. Theorem 6., as the title says: $B(X^{**})$ is the $w^*$-closure of $B(X)$ in $X^{**}$ The proof starts by saying that i) $B(X^{**}$) is $w^*$-...
blomp's user avatar
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Simple way to show that standard basis on $\ell^p$ is weakly pre-compact?

Consider the sequence $e_n = (0,0,\ldots,0,1,0,\ldots)$ in $\ell^1$ which is weakly convergent to zero in $\ell^p$. for all $1\leq p \leq \infty.$ It is then obvious, from the theorem that sequences ...
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Concave conjuguate and weak star topology

I consider a TVS locally convex and separated. I define on it the concave conjuguate of a concave and upper semi continuous function as $$ f^{*}(x^{*}) =\inf_{y\in X}\left\{x^{*}(y) - f(y)\right\},\...
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There exists the converse of this corollary from Brezis?

In Brezis's Functional Analysis, there is a corollary Corollary 3.30. Let $E$ be a separable Banach space and let $\left(f_{n}\right)$ be a bounded sequence in $E^*$. Then there exists a subsequence $...
Francesca's user avatar
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Is the weak topology in separable Sobolev spaces induced by a metric?

I know that the spaces $W^{k,p}(\Omega)$ are separable and reflexive ($\Omega \subset \mathbb{R}^n$ open and bounded) for $p \in (1, \infty)$. I also learned that in a separable Banach space $X$ there ...
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Closed convex hull of a set is the unit ball.

Let $X$ be a (real) Banach space and $A \subseteq X^{*}$ some weak*-compact subset of a unit ball in $X^*$. Furthermore, assume that for any functional $f \in A$, we have $-f \in A$. Now we know that, ...
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Is the weak topology of $\mathbb{C}$ identical to the topology induced by Euclidean distance?

When $X$ is a Banach space, in my textbook, the weak topology is defined as below: For every point $x_0 \in X$, finite subset $F \subset X^\ast$, positive real number $\varepsilon > 0$, we define $...
kuHamrry's user avatar
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Are weak and weak-* topologies translation-invariant?

Let me first fix definitions and notation. Let $X$ be a normed space over $\mathbb{K}$ (either $\mathbb{R}$ or $\mathbb{C}$) and $X^*$ its dual space. Let $$J_X : X\to X^{**}, J_X(x)(f) = f(x)$$ ...
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Consider the weak topology induced by all linear functionals. Is it determined on finite dimensional subspaces?

Let $X$ be a vector space and equip it with the weak topology induced by all linear functionals $X\rightarrow \mathbb{C}$. I am wondering if this topology is so strong that it is determined on finite ...
Jonathan Hole's user avatar
2 votes
1 answer
61 views

How does the weak topology imply the definition of weak convergence?

Let $X$ be a Banach space. Then the weak topology on $X$ is defined as the coarsest topology such that the linear functionals in $X^*$ are continuous. A sequence $x_n \in X$ weakly converges to $x$ if ...
CBBAM's user avatar
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Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?

Assume $(\Omega, \mu)$ is a probability space. Consider a class of functions $S$ all contained in the unit ball of $L^\infty(\Omega, \mu)$. Let $f \in L^\infty(\Omega, \mu)$ be contained in the weak $...
David Gao's user avatar
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4 votes
2 answers
263 views

Definition of weak convergence

Weak convergence came up in my PDE class and I'm trying to understand it even though I lack background in topology and functional analysis. Please check if my understanding is correct. Definition: Let ...
Len's user avatar
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Prove $X$ has the weak topology where $p:X\to Y$ is a covering map.

This question has been answered here, but the method used is different than the one I learned in my algebraic topology course, which I believe I should use. I showed that given that $Y$ is a CW ...
user13121312's user avatar
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Extension of a continuous function $f:X\rightarrow X$ to a function $g: E\rightarrow E$ such that $X$ is embedded in $E$, a Banach space.

We consider a compact metric space $X$ and a continuous function $f:X\rightarrow X$. I know it's possible to embed $X$ into a Banach or separable Hilbert space $E$ using the weak star topology. ...
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Exercise 3.1.23 in Salamon's Functional Analysis

I'm reading the book Functional Analysis written by Dietmar A. Salamon.The following problem is the exercise 3.1.23 in the book. Let X be a Banach space and suppose the dual space of X is separable....
L Zhang's user avatar
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Proof verification regarding weak-continuity

I want to prove that For $V$ a normed space, $f: V \rightarrow \mathbb{K}$ a linear form that is continuous with respect to the weak topology, f is strongly continuous. By definition, the weak ...
Olimani's user avatar
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The map $T: (H^1 (I), \|\cdot\|_{L^2}) \to \mathbb R, u \mapsto \|u\|_{H^1}$ is lower semi-continuous

Let $I$ be the open interval $(0, 1)$. Consider the map $$ T: (H^1 (I), \|\cdot\|_{L^2}) \to \mathbb R, \quad u \mapsto \|u\|_{H^1}. $$ I would like to verify that $T$ is lower semi-continuous (l.s.c.)...
Akira's user avatar
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Is this comment about weak convergence in $W^{1,2}(\Omega)$ correct?

I'm reading @emily20's question in this thread Let $\Omega\subset \mathbb{R}^n$ be open and bounded. Also let $W^{1,2}(\Omega)=\left\{u \in L^{2}(\Omega)|\,\, \forall \alpha \in \mathbb{N}^{n}:|\...
Akira's user avatar
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Existence of a weak-star limit of a family of operators in $B(\mathcal{H})$

Thank you in advance for reading this question, and your thoughts. I am working with a family of operators $(A_{s,\alpha})_{s\geq 0,\alpha>0}$ in the space $B(\mathcal{H})$ (the space of bounded ...
crimsonmist's user avatar
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Exercise 3.19 in Brezis' Functional Analysis

Exercise 3.19: Let $E = \ell^p$ and $F = \ell^q$ with $1 < p < \infty$ and $1 < q < \infty.$ Let $a:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $$|a(t)| \leq C|t|^{\frac{p}...
D4c's user avatar
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TVS Completion of Banach space wrt weak topology

We're talking about weak topologies in my FA courses and I thought of the following question, that I don't know the answer to. Say we have a normed vector space $X$ which we view as a TVS with its ...
ham_ham01's user avatar
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Definition of weak-star topology of bounded linear maps from space to dual of other space

Let $X,Y$ be two Banach spaces. I'm asked to define the weak-star topology in the set of bounded linear maps from $X$ to $Y^{\prime}$, $B(X,Y^{\prime})$, where $Y^{\prime}$ is the dual space of $Y$. ...
BasicUser's user avatar
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1 answer
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Showing the following subset of the dual space is compact

Let $X$ be a real, normed space,and $I$ a non-empty index set. Let $ \{x_i : i\in I\} \subset X$ and $\{ \alpha_i : i\in I\} \subset \mathbb{R}_{\geq 0}$, $R > 0$. For $$ S := \{ f\in X^* : ||f||\...
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Riesz Representation Theorem & Weak closure of orthonormal basis in Hilbert space [closed]

So here it was shown in an infinite dimensional seperable Hilbert space $H$, with the orthonormal basis $E=\{e_1,e_2,\dots \}$ that $0\in \overline{E}^w$, where the right hand side denotes the weak ...
MarvinsSister's user avatar
1 vote
2 answers
180 views

Is the set of probability measures on a closed set closed in the weak topology?

Let $X$ be a metric space and $A\subseteq X$ be a closed subset of $X$. Let $\mathcal{P}(X)$ denote the set of all probability measures on $X$ equipped with the weak topology. Is $\mathcal{P}(A)$ a ...
Trailblazer's user avatar
3 votes
1 answer
80 views

Embedding of $\mathcal{D}(\Omega)$ in $W^{k,p}_0(\Omega)$

$W^{k,p}_0(\Omega)$ is defined as the closure of the set of all $C_c^{\infty}(\Omega)$ under the topology generated by the norm $W^{k,p}(\Omega)$. So clearly the identity map from $\mathcal{D}(\Omega)...
Veronica's user avatar
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Brezis' exercise 8.4: weak convergence of a sequence in $W^{1, p} (\mathbb R)$

I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.4 Fix a function $\varphi \in C^\infty_c (\mathbb R)$ such that $\varphi \neq 0$. Let $u_n (x) := \varphi (x+n)$ ...
Akira's user avatar
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4 votes
2 answers
131 views

Intersection of weak topologies

Let $X$ be a real vector space, let $X^\star$ be its algebraic dual, and fix two vector subspaces $\mathscr{A}, \mathscr{B}\subseteq X^\star$ such that $\mathscr{C}:=\mathscr{A}\cap \mathscr{B} \neq \{...
Paolo Leonetti's user avatar
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Brezis' exercise 8.2.1: a bounded sequence in $W^{1, p}(I)$

Let $I := (0,1)$. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.2.1 Assume that $(u_n)$ is a bounded sequence in $W^{1, p}(I)$ with $1<p \le \infty$. Show ...
Akira's user avatar
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Super confused about the ultra-weak topology on the space of operators on a Hilbert space

Let $H$ be a Hilbert space, let $\mathcal{L}(H)$ denote the bounded linear functions from $H$ to itself and consider the two topologies on $\mathcal{L}(H)$: (1) $\tau_1$ is the topology where a base ...
Charles Ryder's user avatar
3 votes
1 answer
114 views

Ultra-weak topology on space of operators: conditions for convex set to be closed

Notation: Let $H$ be a Hilbert space and let $\mathcal{L}$ denote the set of bounded linear operators from $H$ to itself. On $\mathcal{L}$ we have have the ultra-weak topology which is induced by the ...
Charles Ryder's user avatar
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18 views

Weak-star spanning system in $L^{\infty}(\mathbb{R})$

Recently, I have learned the following surprising result of Hedenhalm and Montes-Rodriguez(2011, Ann. Math. Theorem 3.1): As $n$ ranges over the integers, the functions $e^{\pi inx}$ and $e^{\pi i\...
Tomas's user avatar
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1 vote
1 answer
83 views

Weak and strong relative topologies coincide on compact sets

Let $\left(\mathcal{X}, \tau\right)$ be a locally convex topological space and $K$ a compact subset. Is there a reference for the proof that on $K$ the relative topology coincides with the relative ...
George Gavrilopoulos's user avatar
1 vote
1 answer
122 views

Weak* closure of sets

I struggle with the following exercise about the weak* closure of a set. Weak and Weak* topology is a bit a weak (pun intended) spot of mine so i would like to ask if somebody could take a look, ...
NoIdea's user avatar
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1 vote
1 answer
41 views

Neighbourhoods of $\mathbf{1}$ within the weak topology

Consider a weak topological space. I am trying to show or explain why that for every weak neighbourhood $N$ of $\mathbf{1}$, there is a finite set of vectors such that $N$ contains every vector $\...
Craig Lutic's user avatar
2 votes
1 answer
110 views

Does $x_n \rightharpoonup 0$, where $\|x_n \| \leq 1$ imply $Ax_n\rightarrow 0$, when $A$ is a compact, self-adjoint projection? [duplicate]

Let $(x_n)$ be a sequences in a Hilbert space $H$, with $\|x\|\leq 1$, where $x_n \rightharpoonup 0$. Let $A: H\rightarrow H$ be a linear, compact, self-adjoint projection i.e. $A^2=A$ and $A=A^*$. ...
MackeyTopology's user avatar
1 vote
1 answer
35 views

Is the weak operator topology equal to $\sigma(L(X,Y), X\times Y^*)$?

I'm asking myself if the weak operator topology is equal to the weak topology $\sigma(L(X,Y), X \times Y^{*},b)$ with $$ \begin{align} b:L(X,Y)\times (X \times Y^{*})\to& [0, \infty)\\ (T,(x,y') \...
Davide Modesto's user avatar
7 votes
1 answer
85 views

If $f_n \rightarrow f$ weakly in $L^p$, then $\sqrt{f_n} \rightarrow \sqrt{f}$ weakly in $L^{2p}$?

Suppose $||f_n||_{L^p(\Omega)} \leq C$, where $\Omega$ is a bounded set in $\mathbb{R}^n$. Moreover, $f_n \geq 0$. Using weak compactness, we know that there exists a subsequence $\{f_{n_k} \}$ such ...
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