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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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 ...
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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....
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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 ...
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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.)...
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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}:|\...
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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 ...
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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
$$|a(t)| \leq C|t|^{\frac{p}...
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TVS Completion of Banach space wrt weak topology
We're talking about weak topologies in my FA courses and I thought of the following question, that I don't know the answer to.
Say we have a normed vector space $X$ which we view as a TVS with its ...
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Definition of weak-star topology of bounded linear maps from space to dual of other space
Let $X,Y$ be two Banach spaces. I'm asked to define the weak-star topology in the set of bounded linear maps from $X$ to $Y^{\prime}$, $B(X,Y^{\prime})$, where $Y^{\prime}$ is the dual space of $Y$. ...
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Showing the following subset of the dual space is compact
Let $X$ be a real, normed space,and $I$ a non-empty index set. Let $ \{x_i : i\in I\} \subset X$ and $\{ \alpha_i : i\in I\} \subset \mathbb{R}_{\geq 0}$, $R > 0$. For $$ 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 ...
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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 ...
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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)...
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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 $u_n (x) := \varphi (x+n)$ ...
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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 \{...
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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 ...
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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 ...
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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 ...
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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\...
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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 ...
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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, ...
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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 $\...
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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 $A=A^*$. ...
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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') \...
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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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Weaker norm induces weaker weak topology
Let $(E, \|\cdot \|)$ be a real Banach space. Let $\tau_1$ be the weak topology induced by $\|\cdot\|$. Let $[\cdot]$ be another norm on $E$ that is weaker than $\|\cdot\|$, i.e., there is $C>0$ ...
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Is $\sigma(E, E^*)$ weaker than $\tau$?
Let $(E, \|\cdot \|)$ be a real Banach space. Let $[\cdot]$ be another norm on $E$ that is weaker than $\|\cdot\|$, i.e., there is $C>0$ such that $[u] \le C \|u\|$ for all $u\in E$. Let $\tau$ be ...
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Characterization of weakly compact operators
Let $X$ and $Y$ be normed spaces, I've read that if $T:X^*\to Y$ is a bounded linear operator such that $T^*(Y^*)\subset X$ then $T$ is weak*-to-weak continuous.
First question. Does $T^*(Y^*)\subset ...
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Difference between weak and strong convergence
I don't really understand the difference between weak convergence and strong convergence of operators.
According to my script, strong convergence of a bounded operators $A_n,A$ on a Hilbert space $H$ ...
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Brezis' exercise 6.7.1: a characterization of finite-rank operators
I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
Let $E, F$ be Banach spaces and $T:E \to F$ a bounded linear operator. Consider the following properties:
(P) If $(u_n)$ is a ...
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Every weak neighborhood of $0$ contains a finite co-dimensional subspace
I'm trying to prove a statement mentioned in this answer, i.e.,
In an infinite dimensional normed space $(E, |\cdot|)$, every weak neighborhood of $0$ contains a finite co-dimensional subspace.
...
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Is there a sequence $(x_n) \subset E$ such that $|x_n|=1$ for all $n$ and that $x_n \to 0$ in $\sigma(E, E^*)$?
Let $(E, |\cdot|)$ be an infinite-dimensional Banach space and $\sigma(E, E^*)$ its weak topology.
Is there a sequence $(x_n) \subset E$ such that $|x_n|=1$ for all $n$ and that $x_n \to 0$ in $\...
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On an infinite dimensional locally convex space, no weakly-continuous semi-norm is a norm
Let $X$ be an infinite dimensional locally convex space that separates points and $X^*$ its dual. I would like to prove that no $\sigma(X, X^*)$-continuous semi-norm is actually a norm.
What I have ...
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Probability Notations and Continuity
I started working on probability theory and there is something I am confused with.
Consider a transition function p on a countable and compact space $X$ that is $p(x^\prime|x)$.
How can I understand ...
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Weak continuity for a possibly nonlinear functional on a normed space
I am reading the fifth chapter ("Dual Spaces") from David Luenberger's Optimization by Vector Space Methods (1969). In Section 5.10, weak continuity for possibly nonlinear functionals on ...
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Definition of weak* continuity for possibly nonlinear functionals on normed dual spaces
I am reading the fifth chapter (on Dual Spaces) from David Luenberger's Optimization by Vector Space Methods. In Section 5.10, the author has defined weak continuity for possibly nonlinear functionals ...
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Why is $M$ closed in the weak topology of $V = H^{1,2}(\varOmega;\mathbb{R}^N)$?
Let $\varOmega$ be a bounded domain in $\mathbb{R}^n$, and let $S$ be a
compact subset in $\mathbb{R}^N$. And also let $u_0\in H^{1,2}(\varOmega;\mathbb{R}^N)$
with $u_0(\varOmega)\subset S$ be given. ...
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What is meant by uniform convergence in a general topological vector space?
Let $X$ be a vector space and $Y$ a space of linear functionals on $X$. Reed and Simon give the following definition:
The Mackey topology on $X$ is the topology of uniform convergence on weak* ...
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Containment of kernels of continuous seminorms
Let $X$ be a locally convex space, whose topology is defined by a family of seminorms $\mathcal{P}$.
Fact. For every continuous seminorm $q: X\to \mathbb{F}$ and positive number $\epsilon>0$, the ...
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Proving that a sequence $\{\delta_n\} \in \ell_\infty^*$ has no weak-* convergent subsequence but has a weak-* convergent subnet
This is problem 12 in Reed & Simon's book on functional analysis.
Let $\{\delta_n\}$ be the sequence in $\ell_\infty^*$ such that
$$\delta_n\big(\{c_k\}_{k=1}^\infty\big) = c_n, \quad \forall \{...
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A sufficient condition for tightness of probability measures
For a sequence $\mu_{n}$ of Borel Probability measures, does $\int f\,d\mu_{n}$ converging for all $f\in C_{b}(\Bbb{R})$ imply that $\mu_{n}$ is a tight sequence?
This is pertaining to the question ...
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If $x_n \to x$ weakly then $\exists y_n \in \operatorname{co}\{x_1, \dots, x_n\}$ s.t. $y_n \to x$.
Let $X$ be a normed space and $x_n \to x$ weakly. I can show that there exists $\{y_n\} \subseteq \operatorname{co}\{x_1, x_2, \dots \}$ s.t. $\|y_n -x \| \to 0$ where co is convex hull. To show the ...
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Weak-$\star$ Convergence in $L_1$
Let
$(X,d)$ be a complete separable metric space;
$\mu$ be a Borel probability measure on $X$;
$(f_n),(g_n)\subset L_1(X,\mu)$ be sequences of non-negative uniformly bounded sequences, bounded by $1$,...
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Brezis' exercise 5.25.3
I'm trying to solve an exercise in Brezis' Functional Analysis
Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $K$ be a non-empty closed convex ...
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Brezis' exercise 5.22.1: non-linear contraction
I'm trying to solve an exercise in Brezis' Functional Analysis
Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $C$ be a non-empty closed convex ...
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Brezis' exercise 5.20.3: an approach based on weak topology
I'm trying to fill in the details of the proof of part (3.) of of exercise 5.20 in Brezis' Functional Analysis.
Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its ...
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Brezis' exercise 5.19
I'm trying to solve below exercise in Brezis' Functional Analysis
Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $u_n, u \in H$ such that $u_n \to ...
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Fix $f \in H$ and let $u_n$ be the projection of $f$ onto $K_n$. Prove that $(u_n)$ converges in $H$
Let $(H, \langle \cdot, \cdot\rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $(K_n)$ be a non-increasing sequence of closed convex subsets of $H$ such that $K := \bigcap_n K_n \...
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Is the inner product $\langle \cdot, \cdot\rangle$ lower semi-continuous w.r.t. the product of weak topologies?
Let $(H, \langle \cdot, \cdot\rangle)$ be a real Hilbert space. Let $\sigma (H, H^*)$ be the weak topology of $H$. Let $\tau :=\sigma (H, H^*) \otimes \sigma (H, H^*)$ be the product topology of $\...
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