All Questions
Tagged with fa.functional-analysis topological-vector-spaces
213 questions
1
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1
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232
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An approximation property in a separable topological vector space
Let $X$ be a topological vector space.
Let us say that $X$ enjoys sequential separablity if there exists a sequence $\{x_n\}$ in $X$ such that for every $x\in X$ there exists a subsequence of $\{...
- 4,058
2
votes
1
answer
297
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Predual theorem proof in Takesaki's volume I
Consider the following fragment from Takesaki's book "Theory of operator algebra I" (Section III.3 ,p133-134).
Why is the boxed line true? I can see that $\epsilon: \widetilde{A}\to A$ is ...
- 175
0
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1
answer
231
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Borel sigma algebra coming from the weak topology on TVS
Let $(X,\tau)$ be a topological vector space. Suppose that, there is a sequence of subsets $X_n\subseteq X$ with,
For every $n\in \mathbb{N}$, the topology $\tau$ on $X_n$ is second countable and ...
- 4,058
3
votes
1
answer
156
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$\varepsilon$-product in Bierstedt's paper
I am reading K.D.Bierstedt's paper Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. I. Journal für die reine und angewandte Mathematik 259 (1973): 186-210. It is ...
- 7,426
5
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2
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359
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Product of inductive limit topologies on $C_c(X)\times C_c(X)$
I have a stupid question about a topology on $C_c(X)$. Here $X$ is locally compact Hausdorff. Can assume $\sigma$-compact if it helps.
Definition (topology on $C_c(X)$): For each compact $K \subset X$,...
- 675
3
votes
0
answers
120
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Approximation of a linear functional by linear continuous functionals
Let $X$ be a locally convex space, $T$ a balanced convex compact set in $X$, and $f:X\to\mathbb{C}$ a linear functional which is (not necessarily continuous on $X$, but) continuous on $T$. It is not ...
- 7,426
5
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1
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2k
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Definition of infinite-dimensional Gaussian random variable
For infinite-dimensional Gaussian measures, we often see the definition of Gaussian random variables like this:
Let $H(\Omega;\mathbb{R})$ be a separable Hilbert space. A random
variable $u \in H$ is ...
- 227
6
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0
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1k
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Condensed/liquid vector spaces and path integrals
[Edited to take into account comments.]
Background
One approach to the problem of making rigorous various measures on spaces of paths (for example, the Wiener or Feynman measure) is the time-slicing ...
- 61
1
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0
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50
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Nested nets of closed bounded star-shaped sets in a semi-reflexive space
Among Hausdorff locally convex spaces, semi-reflexivity is characterized by the weak topology having the Heine-Borel property. It follows that, in a semi-reflexive space, every nested net of closed ...
- 11
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1
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188
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Uniqueness of the predual of a W*-algebra
Consider the following fragments in Takesaki's "Theory of operator algebras" (volume I):
Question: So, we have an abstract Banach space $F$ with $A \cong F^*$. In Lemma 3.6, one considers ...
- 175
2
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1
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111
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About $\sigma$ strong$^*$-functionals and seminorms
I'm reading the book "Theory of operator algebras" by Takesaki. In this book, the $\sigma$-strong$^*$ topology on the space $B(H)$ (bounded operators on the Hilbert space $H$) is defined (...
- 175
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1
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536
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About the normability of the space of continuous functions
Let $A$ be a subset of $\mathbb{R}^n$, and denote by $C(A)$ the space of complex-valued continuous functions defined on $A$. We know that if $A$ is compact then we can define a norm on $C(A)$ so that ...
- 1,173
3
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1
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475
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Strict topology on the multiplier algebra
Let $A$ be a $C^*$-algebra. Let $M(A)$ be its multiplier $C^*$-algebras. The strict topology on $M(A)$ is given by
$$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to ...
- 175
2
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0
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211
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Effect of dualization of density
Let $D\subset X$ be a dense subset of a complete separable locally convex space $X$ over $\mathbb{R}$. Though the question seems simple enough, I can't seem to find the answer in the literature:
If $...
- 109
6
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1
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455
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Is the tensor product of distributions a continuous bilinear map with respect to the weak topology?
Let $X$ and $Y$ be smooth manifolds. The map $\mathcal{D}'(X)\times\mathcal{D}'(Y)\to\mathcal{D}'(X\times Y)$ given by $(S,T)\mapsto S\boxtimes T$ is continuous with respect to the strong topology. Is ...
- 387
2
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1
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189
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Biorthogonal weakly null basic sequences
Let $E$ be a Banach space, let $e_{n}\in E$ and $g_{n}\in E^{*}$ be biorthogonal basic sequences (i.e. $\left<e_n,g_m\right>=\delta_{mn}$ ). Moreover, both of these sequences are weakly null. (...
- 5,529
5
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1
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144
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The tensor product of two topological complexes with closed range
A Künneth formula by Grothendieck/Schwartz states the following:
Let $A, B$ be chain complexes of nuclear Fréchet spaces. If the differentials $d_A, d_B$ are topological homomorphisms (meaning in ...
user126256
1
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1
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144
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What's the size of non standard monad for weak topology?
There have been several works characterizing weak topology by nonstandard analysis, which give rise to the following monad ($X$ is a Hilbert space):
$$\mu(0) = \{y\in{}^{*}X: \forall x\in X ~~ \...
- 75
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1
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172
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Is $C_0(X) $ Frechet-Urysohn with respect to the compact-open topology?
Recall that a topological space is called Frechet-Urysohn if the operations of closure and sequential closure coincide.
Let $X$ be a locally compact Hausdorff space. It is known that $C(X)$ is not ...
- 5,529
0
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2
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123
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$(ST)_{[13]}= S_{[13]}T_{[13]}$ for $S,T \in B(\mathcal{H}\otimes \mathcal{H}).$
Let $T\in B(\mathcal{H} \otimes \mathcal{H})$ where $\mathcal{H}$ is a Hilbert space. We can define operators
$$T_{[12]}= T \otimes 1;\quad T_{[23]}= 1 \otimes T$$
and if $\Sigma: \mathcal{H} \otimes \...
user167952
5
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2
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247
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Is there a topology that makes every basic sequence null?
Let $E$ be a Banach space. Let $F$ be the collection of all $f\in E^*$ such that $\left<f,e_n\right>\to 0$, for every normalized basic sequence $\{e_n\}$. It is easy to see that $F$ is a closed ...
- 5,529
5
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1
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450
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A "proof" that all separately continuous maps on LF-spaces are continuous
Problem
Consider the locally convex spaces $C^\infty(\mathbb{R})$ and $C^\infty_c(\mathbb{R})$, the former equipped with its standard Fréchet topology, the latter equipped with the inductive limit ...
user126256
5
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1
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421
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Is the filtered colimit topology on the space of signed Radon measures linear and locally convex?
Let $X$ be a compact Hausdorff space. In chapter 3 of Peter Scholze's Lectures on Analytic Geometry he considers the space of signed Radon measures on $X$ equipped with the filtered colimit (aka ...
- 153
4
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1
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510
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Strict topology and $*$-strong toppology on $B(H)$ coincide
In the paper Woronowicz - $C^*$-algebras generated by unbounded elements, I read that the $*$-strong operator topology on $B(H)$ and the strict topology on $B(H)$ coincide. I believe this means the ...
user167952
4
votes
2
answers
769
views
smooth functions on closed intervals with values in infinite-dimensional spaces
There are three ways to define when a ($\mathbb{R}$-valued) function on a closed interval is smooth:
$f$ can be extended to a smooth function on $(a - \epsilon, b + \epsilon)$ for some $\epsilon > ...
- 1,141
6
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1
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316
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Compatibility of inductive and projective limits with dualization in functional analysis
Assume $(A_i)_{i \in I}$ is a family of locally convex topological
vector spaces which are all moreover assumed to be Banach spaces.
We assume moreover that $(A_i)_{i \in I}$ has additional
structure ...
- 6,018
4
votes
2
answers
263
views
Sufficent condition for strict morphism of normed vector spaces
Let $K$ be a non-archimedean field of char 0 and a morphism $f:V \rightarrow W$ of normed $K$-vector spaces given. The map $f$ is said to be strict if $V/\ker(f)$ with the quotient topology is ...
- 473
4
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1
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315
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Convenient vector space and its locally convex structure
I'm trying to understand convenient vector spaces, but I'm unsure about the definition of the topology on smooth maps.
A map $f : E \rightarrow F$ between locally convex vector spaces $E$ and $F$ is ...
- 143
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1
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86
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Are bounded sets in second duals of locally convex spaces weak* pre-compact?
Let $X$ be a locally convex Hausdorff space. Then $X$ injects into $X^{**}$ via the canonical map $\kappa: X\to X^{**}$. Now, $X^{**}$ carries the weak* topology. Let $B$ be a bounded set in $X$. Is $\...
- 1
3
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0
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84
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"Weakly" nuclear operators (terminology)
Recently, I'd come across the following kind of operators and I wonder if they have been considered before and given a name.
Let's say that a linear map $T:V\to W$ between locally convex topological ...
- 81
3
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0
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67
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Non-linear weak*-continuous left inverses
Let $T\colon X\to Y$ be a continuous linear surjection between Frechet spaces. Then it is possible to use Michael's selection theorem to show that there exists a continuous (non-linear) function $g\...
- 31
14
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0
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860
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strong topologies on $C_c^\infty$
UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
- 486
6
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2
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509
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A question on Grothendieck space
A Banach space $X$ is said to be Grothendieck if the weak and the weak* convergence of sequences in $X^{*}$ coincide. I have the following two questions.
Question 1. A Banach space $X$ is Grothendieck ...
- 3,343
7
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1
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754
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Closed convex hull in infinite dimensions vs. continuous convex combinations
tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$?
I am essentially asking for the most general, infinite-dimensional analogue of ...
- 71
2
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1
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70
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Equicontinuity-like property of a convex compact set
Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$.
Is there an ...
- 5,529
0
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0
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45
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Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
- 5,405
2
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0
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1k
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Bounded weak and weak-$\star$ topologies and metrics
Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $\overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$). Set
$$
d(x,y) := \sum_{n=1}^\infty 2^{-n} |(x-y, h_n)|...
- 349
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0
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101
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Can a quotient space of a locally convex space have finer topology that its domain?
The following is related to this post.
Let $X=X'$, as sets, and let $T:X \rightarrow X'$ be a surjective map from a countably infinite-dimensional LCS $X$ to itself and equip $X'$ with the final ...
- 5,405
5
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1
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219
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Are linear continuous mappings open on totally bounded sets?
Let $X$ and $Y$ be locally convex spaces, and $\varphi: X\to Y$ a linear continuous mapping. Suppose first that $S$ is a compact set in $X$. Then $\varphi$, being considered as a mapping from $S$ to $\...
- 7,426
1
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48
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Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?
Let $X$ be a Banach space equipped with the Mackey topology $m(X,X^*)$. We suppose that $\big(X,m(X,X^*)\big)$ is separable space. Let $H$ be a countable, $m(X,X^*)$-dense subset with $(H=-H)$.
Let $...
- 117
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4
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796
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On Köthe sequence spaces
I asked this a week ago at math.stackexchange, but without success.
As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his "...
- 7,426
6
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1
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117
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Infra-Pták space that is not Pták
From reading the literature of the 1970s heyday of locally convex spaces, it seems that it was an important open question whether there is an infra-Pták (i.e. $B_r$-complete) space that is not Pták (i....
- 3,816
7
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1
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209
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$GL_1(\mathcal{E}'(\mathbb{R}))$ open in $\mathcal{E}'(\mathbb{R})$?
Let $\mathcal{E}'(\mathbb{R})$ be algebra of all compactly supported distributions on $\mathbb{R}$, equipped with the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$, and with the usual ...
- 73
12
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1
answer
908
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Equivalence of σ-convex hull and closed convex hull
Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as
$$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\,...
- 123
4
votes
2
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1k
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A topological vector space $X$ is separable if its dual space $X^*$ is separable?
Let $(X,\tau)$ be a topological vector space such that the associated dual space $X^*$ is separable. Can we say that $X$ is separable ?
I know that this property is valid for Banach spaces but for ...
- 57
2
votes
1
answer
101
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Sobolev topology on essentially compactly supported Sobolev-"functions"
The locally convex space of essentially compactly-supported $p$-integrable "functions" $\operatorname{L}_{\mathrm{comp}}^p(\mathbb{R}^d,\mathbb{R})$ is defined as the set
$$
\bigcup_{n \in \mathbb{N}} ...
- 5,405
8
votes
2
answers
385
views
Metrizability of a topological vector space where every sequence can be made to converge to zero
This is a follow-up to this answer.
If $E$ is a (real or complex) topological vector space, we say that a sequence $\{x_n\}_{n=1}^\infty$ in $E$ can be made to converge to zero if there exists a ...
0
votes
0
answers
46
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Show that $\big(s(. |C_n)\big)_n$ is equicontinuous on $X^*$
Let $(X,\|.\|)$ be a separable Banach space with dual $X^*$. $\mathcal{P}_{wkc}(X)$ be the class of nonempty, weakly-compact and convex subsets of $X$. For any $C\in\mathcal{P}_{wkc}(X)$ we define ...
- 1
1
vote
1
answer
203
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Continuous function on colimit
Let $X$ be a Banach space and $f:X\rightarrow \mathbb{R}$ be continuous. Suppose that $\{X_n\}_{n \in \mathbb{N}}$ is a strictly nested sequence of sub-Banach spaces, for which $\cup_{n \in \mathbb{N}...
- 5,405
4
votes
1
answer
574
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Criterion for weak convergence of sequences
Let $E$ be a normed space and let $F\subset E^{*}$. It is known that $F$ is dense if and only if the restriction of $\sigma(E,F)$ on $B_E$ coincides with the weak topology.
Hence, if $F$ is dense and ...
- 5,529