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Questions tagged [locally-convex-spaces]

For questions about topological vector spaces whose topology is locally convex, that is, there is a basis of neighborhoods of the origin which consists of convex open sets. This tag has to be used with (topological-vector-spaces) and often with (functional-analysis).

7
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0answers
261 views

If locally convex topologies exhibit the same dual spaces, do they exhibit the same continuous linear operators?

Consider the following setting: Let $X, Y$ be vector spaces over the field $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$. Furthermore, let $\tau_1, \tau_2$ be locally convex topologies on $X$ and $\...
2
votes
1answer
302 views

Linear Map is injective iff Adjoint has dense image

$A,B$ are locally convex topological spaces, $f:A\to B$ is linear, continuous. I want to verify the adjoint map has dense image in $A^*$ with respect to the weak* topology if and only if $f$ is ...
2
votes
0answers
53 views

Why is this set $Z$ convex?

I am reading the book Methods of Modern Mathematical Physics by Reed & Simon. In one of their lemma they claim that for a locally convex space $X$, Let $V$ be an open, convex, balanced subsets ...
1
vote
1answer
152 views

First countability, nets and sequences in the weak topology.

According to this book (see print below) the weak topology (as well as the weak* topology) is first countable. However the weak topology should be first countable only in the finite dimensional case. ...
1
vote
1answer
176 views

Why sequential continuity from $E$ to $E'$ implies continuity?

Let $(E,\|\cdot\|)$ be a separable Banach space. Let $E'$ be the topological dual of $E$ equipped with the weak* topology $w^*$. I read that a certain linear operator $J:(E,\|\cdot\|)\to (E',w^*)$ is ...
1
vote
1answer
89 views

Is a sequentially continuous map $f:E'\to E'$ continuous?

I have read that for a separable complete locally convex space $E$, any sequentially continuous linear map $f:E'\to \mathbb{K}$ is continuous (where $E'$ is equipped with the weak*-topology). Is ...
2
votes
0answers
135 views

Inductive limits of test function spaces and local convexity

This question is motivated by the test function spaces for distributions, but to be simple just let $K_i=[-i,i]\subseteq\mathbb R$, $i=1,2,\ldots$, and consider the inductive limit topology of the ...
2
votes
0answers
40 views

Non-comparability of two particular polar topologies

Is there an example of a locally convex Hausdorff space $(X, \tau)$ such that on its dual $X'$ the topology $\tau_{c_0}$ of uniform convergence on $\tau$-null sequences and the topology $\tau_c$ of $\...
1
vote
0answers
46 views

Question related to Fréchet space

I would like to know if the following property that is true for Banach spaces, it holds for Fréchet space as well. The question is: Let $X$ be a Banach reflexive space and $Y$ be a Fréchet space. ...
0
votes
1answer
40 views

Possible explanation for a terminology confusion

In the usual sense, given two spaces $X,Y$ and a family of functions $F$ which map $X$ into $Y$, we say $F$ separates points if for any distinct $x_{1,2}\in X$ there exists at least one $f\in F$ such ...
3
votes
0answers
45 views

Is $A = \{ (z_1 , z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2|\}$ an absorbing, balanced and convex set?

Is $A = \{ (z_1 , z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2|\}$ an absorbing, balanced and convex set? I am new in StackExchange I from Colombia, because I don't write English very well. Edited from ...
0
votes
1answer
113 views

Infinite-dimensional spaces for which strong and weak topologies coincide

For a dual pair $\langle X, Y \rangle$ of vector spaces $X$ and $Y$ denote by $\sigma(X,Y)$ the weak topology and by $\beta(X,Y)$ the strong topology on $X$. Consider the following known statement (S)...
2
votes
0answers
48 views

Relations between different definitions of polar topologies

Let $\langle X, Y \rangle$ be a separated dual pair of (real) vector spaces. A non-empty family $\mathcal{A}$ of non-empty subsets of $Y$ is called polar if every $A \in \mathcal{A}$ is bounded (w.r....
1
vote
1answer
87 views

Approximation property for $C^k([0,1]^m)$

This must be well-known, of course, so excuse me my ignorance. I think, the Banach space $C^k([0,1]^m)$ (of $k$ times smooth functions on a hypercube $[0,1]^m$) must have the approximation property. ...
2
votes
1answer
125 views

Mackey topology vs. uniform convergence on weakly compact convex sets

Let $X$ be a locally convex Hausdorff space and $X'$ its dual space. By the Mackey-Arens theorem, there is a finest locally convex topology $\tau$ on $Y$ such that $(Y, \tau)' = X$. $\tau$ is called ...
2
votes
0answers
1k views

How do I prove $\{ x : Ax=b \}$ is convex?

I have a constraint $Ax=b$, where $A$ is $m\times n$, b is $m\times1$. I am wondering how should I prove that $\{ x \in \mathbb{R}^n \mid Ax=b \}$ is convex. Attempt: $\{x \in \mathbb{R}^n \mid Ax=...
3
votes
0answers
219 views

The Minkowski gauge is always bounded by some semi-norm?

My problem sheet asks the following: Let $X$ be a locally convex space generated by the semi-norms $(p_i)_{i\in I}$. Let $A$ be some convex neighborhood of $0$, and $p$ the Minkowski gauge of $p$. ...
0
votes
1answer
314 views

Hessian Matrix Principal Minor and Convexity

Say I have a Hessian Matrix: $\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}$ The principal minor for 2x2 symmetric matrix are: $D_1=0$, $D_2=ac-b^2$ which are: $D_1=0, D_2=-1$. ...
3
votes
1answer
86 views

Dense locally convex subspace

$V$ is a locally convex space. I can't manage to prove that a subspace $M$ is necessarily dense if $\forall f\in V^*(f(M)=\lbrace{0\rbrace} \implies f(V)=\lbrace{0\rbrace})$. All I have is $\exists ...
1
vote
1answer
18 views

Boundedness in locally convex space

Consider a vector space $X$ equipped with a separating countable family of seminorms $(p_i)_{i\in\mathbb{N}}$. I'm told that the boundedness of a set in $X$ with respect to each one of the seminorms ...
0
votes
1answer
126 views

Continuous Map for the Compact Open Topology

Suppose I have a map $\Phi:\mathbb{R}\rightarrow C_b(\Omega)$ where the right hand side is equipped with the compact open topology $\tau_{co}$. How to show that such a function is continuous with ...
4
votes
0answers
115 views

State space of $C([0,1])$

Consider the $C^*$-algebra $A := C(X)$, where $X$ is a compact Hausdorff space. Denote by $S(A)$ the state space of $A$. In the weak$^*$ topology this is a convex compact set such that $S(A)$ is the ...
1
vote
0answers
46 views

Locally Convex Topology on $C_b(\Omega)$

Let $\Omega$ be a locally compact Hausdorff space and look at $C_b(\Omega)$. Then we can define a topology $\tau$ on this space by being the finest locally convex topology agreeing with the compact ...
2
votes
1answer
214 views

Continuous seminorms

If I have a locally convex vector space $S$ equipped with a countable family $(p_i)_{i \in I}$ of seminorms, is it correct that the topology remains unchanged if I add an extra seminorm $q$ satisfying ...
1
vote
0answers
322 views

Metrizability of space test $\mathcal{D}(\Omega)$, and of distributions spaces $\mathcal{D}'(\Omega)$

Since each open set $\Omega \subset \mathbb{R}^n$ admits a increasing exhaustion of compact set $K \subset \Omega$, then the space of test functions are rewritten as a countable union $\mathcal{D}(\...
3
votes
1answer
112 views

Proof that a locally convex space $E$ is regular

Where a topological space $E$ is a regular space if, given any closed set $F$ and any point $x$ that does not belong to $F$, there exists a neighbourhood $U$ of $x$ and a neighbourhood $V$ of $F$ that ...
3
votes
5answers
605 views

(Hausdorff ) Locally convex spaces and their “natural” metric

Today we were introduced to locally convex spaces, defined thusly: A vector space is locally convex iff it has a family of semi-norms $(p_i)$ such that $x=0$ if and only if $p_i(x)=0$ for all $i$. ...
0
votes
1answer
169 views

Extension of continuous linear functionals from a closed subspace to the whole locally-convex space

Consider a Hausdorff, real (but I believe this not to be relevant), locally-convex topological linear space $X$ and a closed linear subspace $Y$. Let $f \in Y^*$ (the topological dual). I want to show ...
0
votes
1answer
354 views

Proof of the Banach-Alaoglu theorem

I have some doubts about the point of this proof of the Banach-Alaouglu theorem: "The cloesed ball $B_{E'}:=\lbrace u \in E' : \left \| u \right \|_{E'} \leq 1 \rbrace$ of dual $E'$ of a normed space ...
2
votes
2answers
155 views

Are closed subspaces of reflexive locally convex spaces reflexive?

We know that if $X$ is a Banach space which is reflexive, then any closed subspace of $X$ is reflexive, could we extend the conclusion to any locally convex topological vector space, where $X^{'},X^{''...
2
votes
1answer
154 views

Continuous semi-norms on subspace

Suppose $X$ is a locally convex topological vector space, let $P$ be the set of all continuous semi-norms on $X$. Suppose $M$ is a subspace of $X$, denote $P|_M$ as the set of semi-norms in $P$ ...
1
vote
0answers
102 views

Show that a differentiable function on a convex space is injective

Let $n \in \Bbb N$ and $G \subset \Bbb R^n$ be a convex space, $f: G \to \Bbb R^n$ continuously differetiable and $$\det\begin{pmatrix} \frac{\partial f_1}{\partial x_1}(c_1)& \cdots &\frac{\...
2
votes
0answers
50 views

Is strong operator topology space $(B(H), SOT)$ reflexive?

It is true that $(B(H), SOT)$ is semireflexive, in which $H$ is a Hilbert space, and $B(H)$ is the set of all bounded linear operators from $H$ to $H$ with strong operator topology. As a starting ...
2
votes
2answers
120 views

A doubt about the vectorial topology on $\mathcal{D}(\Omega)$

We denote with $\mathcal{U}_0$ the family of all subsets $U \in \mathcal{D}(\Omega)$ convex and balanced such that $U \cap \mathcal{D}_K(\Omega) \in \mathcal{T}_K$, where $\mathcal{T}_K$ is the ...
2
votes
0answers
243 views

Is it true that the closed convex hull of a compact subset of the dual equipped with the w*-topology is compact?

Let $X$ be a Banach space. Consider the dual $X^*$ equipped with the weak*-topology. Is it true that the closed convex hull of a compact subset $K$ of the dual $X^*$ is compact? ps: I know that the ...
9
votes
2answers
873 views

Definition of the convolution with tempered distributions and Schwartz function

In the book where I'm studying there is the following exercise. If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=...
0
votes
1answer
351 views

Dual space $E'$ is metrizable iff $E$ has a countable basis

I saw that it was already asked, but the book where I'm studying is slightly different. Recall some definition, if $E$ it's $\mathbb{K}$-vector space and let $\mathcal{E}$ be a vector subspace of the ...
0
votes
1answer
58 views

discrete convexity arising in a simple discrete optimization problem

Let $S$ be a fixed integer satisfying $S \ge 1$, let $a$ range over the integers between $1$ and $S$ inclusive, and for $i = 1, \dotsc, a$, let each $x_i$ range over the nonnegative integers, such ...
0
votes
0answers
79 views

Metrisability of locally convex spaces/weak topology

The weak topology of a Banach space $X$ is the locally convex topology associated to the family of semi-norms $$ p_f(x)= |f(x)|, \qquad f\in X^*. $$ If $X^*$ is separable, it obviously suffices to ...
2
votes
2answers
343 views

A characterization of tempered distributions

The Schwartz space on $\mathbb{R}^n$ is the function space $$ S \left(\mathbf{R}^n\right) = \left \{ f \in C^\infty(\mathbf{R}^n) : \|f\|_{\alpha,\beta} < \infty,\, \forall \alpha, \beta\in\mathbb{...
0
votes
1answer
59 views

can I get weak convergence in sobolev spaces from convergence of distributions

my question is the following. Given a sequence $(f_k)_k$ in $W^{1,q}(\Omega)$ with $q \in (1,\infty)$ and $\Omega \subseteq \mathbb{R}^n$ open and bounded. If I want to show $f_k$ converges in $W^{1,...
3
votes
2answers
631 views

What is Convex about Locally Convex Spaces?

This might be a silly question, but what motivates the name "locally convex" for locally convex spaces? The definition in terms of semi-norms seems to have nothing to do with convexity or with the ...
2
votes
0answers
152 views

Day's fixed point theorem

Day's fixed point theorem (Theorem 1.3.1; Lecture on amenability; Volker Runde) Let $G$ be a locally compact group. The following are equivalent: $G$ is amenable. If $G$ acts (from left side) ...
3
votes
2answers
305 views

Product topology on a product space of normed spaces is normable iff the product is finite [duplicate]

Suppose $(X_{i}, \Vert \cdot \Vert_i)_{i\in I}$ are all normed spaces over the same field $\Phi= \mathbb{R}, \mathbb{C}$ and suppose $X= \prod_{i \in I} X_i$ is the product space. I want to show that ...
7
votes
1answer
159 views

Does the operation of completion preserve injectivity?

It seems to me I saw a counterexample somewhere, but I can't find it, can anybody help me? Let $\varphi:X\to Y$ be a linear continuous map of locally convex spaces, and $\widetilde{\varphi}:\...
1
vote
0answers
191 views

Why is $x\in X$ a weak star continuous linear functional in the dual?

I am reading an excerpt from Infinite Dimensional Analysis by Aliprantis and on page 235 it claims that if $X$ is a normed space, then "$x$ is a weak* continuous linear functional by definition". ...
0
votes
1answer
110 views

Proof of property of unique nearest point equal to convex.

In the picture below, I really don't know how to use the compactness to get there will be a ball $C$ with maximal radius. I report this book for my classmate and teacher. I tell they the $r$ has a ...
1
vote
0answers
41 views

connect and arcwise-connect in locally convex space

Let X be a locally convex vector space and let G be an open connected subnet of X. How to show that G is arcwise-connected? I only can show that G is path-connected but do not know why G is arcwise-...
1
vote
0answers
60 views

Do these Topologies define the same open sets?

I am trying to understand weak Topologies by reading John Conway's Course in Functional Analysis and he lists a bunch of theorems such as: "If $X$ is LCS, $(X,wk)^{*} =X^{*}$" which are getting very ...
0
votes
1answer
42 views

Weak and weak* topologies

If X is a locally convex vectorspace, does the weak and weak* topologies on X* coinside? If so how to prove it?