Skip to main content

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).

Filter by
Sorted by
Tagged with
2 votes
1 answer
35 views

Proof that convergence in probability is not locally convex

On Wikipedia, it is stated that the topology induced by convergence in probability is not locally convex. However, no proof nor reference is provided, and I couldn't find anything on Google. I think ...
kalkuluss's user avatar
0 votes
1 answer
36 views

Convex hull of bounded set is bounded. Is my proof right?

I want to prove the following theorem. Let $X$ be a topological vector space. If $X$ is locally convex then the convex hull of every bounded set is bounded. My proof: If $A$ is bounded, for every ...
xyz's user avatar
  • 709
2 votes
0 answers
50 views

Vector space topologies stronger than the strongest locally convex topology

Every real vector space has the strongest locally convex topology, which is the topology generated by all the convex sets whose intersection with every line is an open interval. What about topologies ...
Alexey's user avatar
  • 2,210
1 vote
1 answer
40 views

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
0 votes
1 answer
19 views

Do "halves" of open sets exist in locally convex vector spaces?

Let $V$ be a locally convex Hausdorff topological vector space (over $\mathbb{R}$) and let $U\subseteq V$ be an open neighbourhood of the origin. Does there always exist another open neighbourhood $U'$...
Hans's user avatar
  • 3,615
-1 votes
1 answer
49 views

convex combination of probability measures [closed]

$\left( \Omega,\mathcal{A} \right)$ is a measurable space and $\mu,\nu$ are probability measures on it. Prove any convex combination of $\mu$ and $\nu$ is also a probability measure on this space. ...
tom31415's user avatar
2 votes
0 answers
54 views

Why is closedness crucial for a barrel set to be a neighborhood of the origin in a Banach space

Apology for asking highly related but subtly different questions within a very short amount of time. Let $X$ be a real Banach space, and $A\subset X$ be balanced, convex, and absorbing. Two facts are ...
user760's user avatar
  • 1,670
0 votes
0 answers
36 views

Minkowski functional as a norm

Related to my previous question. Let $X$ be a real Banach space, and $A\subset X$ a balanced, convex, absorbing set that is bounded. Then $(X, p_A)$, where $p_A$ is the minkowski functional, is a ...
user760's user avatar
  • 1,670
1 vote
0 answers
25 views

Schauder bases in inductive limits

Assume $X_n$ is a family of nested Banach spaces (i.e. $X_n\subset X_m$ whenever $m>n$ and the inclusion map is continuous) and denote by $X$ the inductive limit $\lim X_n$. Assume moreover that $X$...
Pelota's user avatar
  • 1,098
3 votes
1 answer
49 views

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 \...
Peter's user avatar
  • 476
3 votes
0 answers
15 views

Complete locally convex topological vector spaces are not stable under extension

I've heard that complete locally convex topological vector spaces are not stable under extension. However, I don't know of any example. What would be an example of a complete topological vector space $...
Smiley1000's user avatar
  • 1,649
1 vote
2 answers
38 views

Question about weak and pointwise convergence

I have a question about weak topology. Definition: If $X$ is a LCS, the weak topology on $X$, denoted by "wk" or $\sigma(X,X^*)$, is the topology defined by the family of seminorms $\{p_f : ...
Peter's user avatar
  • 476
0 votes
0 answers
20 views

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 ...
Peter's user avatar
  • 476
0 votes
1 answer
30 views

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
  • 476
2 votes
1 answer
41 views

Lemma about Minkowski Functional in topological vector spaces

I am trying to prove the following lemma: Suppose $X$ is a topological vector space. Show that if $S \subseteq X$ is a convex (open) neighborhood of $0$ there exists a non-negative continuous ...
Philip's user avatar
  • 635
3 votes
0 answers
52 views

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
  • 476
3 votes
1 answer
68 views

Is every extreme point in a compact convex set contained in a defining supporting hyperplane?

Let $K \subseteq X$ be a compact convex subset of a locally convex space $X$. Let $k \in K$ be an extreme point. Question 1: Does there exist a supporting hyperplane of $X$ containing $k$? I think the ...
tcamps's user avatar
  • 6,033
2 votes
0 answers
55 views

$\ell^p(\mathbb{N})$ for $0 < p < 1$ is not locally convex (seminorms definition) [duplicate]

I am reading Conways A Course in Functional Analysis. A locally convex space is a topological vector space whose topology is defined by a family of seminorms $\mathcal{P}$ such that $\bigcap_{p \in P} ...
Philip's user avatar
  • 635
2 votes
2 answers
53 views

If $L$ is an interval and $C$ is convex, is $LC$ convex too?

Let $LC = \{\lambda v \mid \lambda \in L, v \in C\},$ where $C$ is some convex set of the vector space $V.$ If $L$ is an interval, is $LC$ convex? I thought and assumed so for a while, and some ...
William M.'s user avatar
  • 7,706
2 votes
2 answers
97 views

Condition for showing families of seminorms generate same topology

There is a statement about locally convex spaces in Reed & Simon, Methods of Modern Mathematical Physics (Vol I, Section V.1) that is given without a proof. The statement is: Given two families of ...
Mark's user avatar
  • 148
0 votes
0 answers
28 views

Maximizing a Function with Cosine Terms and Scalar Parameters

I am facing a challenging maximization problem involving a function with cosine terms and scalar parameters, and I would appreciate some insights or guidance on how to approach it effectively. $$\max_{...
ANAS.C's user avatar
  • 103
3 votes
1 answer
43 views

Final topology coinduced by topological vector spaces

Let $X$ be a vector space, $(X_i)$ a collection of (not necessarily locally convex) topological vector spaces, and $T_i \colon X_i \to X$ linear maps. Then the $T_i$ coinduce a final topology $\...
Danny's user avatar
  • 636
0 votes
1 answer
47 views

Example of a LCS with a countably compact barrel

I am self-studying topological vector spaces and I wonder if there is an example of a sequentially complete LCS with a countably compact barrel. I am a complete beginner and really can't think of any. ...
somegirl's user avatar
  • 105
3 votes
2 answers
343 views

Motivation behind locally convex spaces, seminorms, and Frechet spaces

I am looking for some motivation behind the definition of locally convex spaces, seminorms, and Frechet spaces. Since all three concepts are related I have grouped them as one question. I am familiar ...
CBBAM's user avatar
  • 6,295
1 vote
0 answers
42 views

Metrizability of topology generated by a countable family of semi-norms

I have the following task. Let $X$ be a vector space equipped with a family $\mathcal{P} = \{\rho_{n}\}_{n\in \mathbb{N}}$ of semi-norms. I want to prove that: This family of semi-norms generate a ...
InMathweTrust's user avatar
1 vote
0 answers
75 views

On relatively compact sets in the space of holomorphic functions

I'm having trouble with an exercise about holomorhpic functions. Let $\Omega \subset \mathbb{C}$ be an open convex set and let $F \subset H(\Omega)$ be a relatively compact set in the open-compact ...
Contrad's user avatar
  • 47
0 votes
3 answers
73 views

A line segment in the closure of a convex subset of a topological vector space

I am trying to solve the following problem. Let $C$ be a convex subset of a topological vector space $X$. Let x be in the interior $C^\circ$ and let y be in the closure $\bar{C}$. I am asked to prove ...
liyiontheway's user avatar
1 vote
1 answer
42 views

Continuity wrt locally convex topology implies continuity wrt to at least one generating seminorm?

Let $(V,\mathcal{P})$ be a locally convex topological vector space, that is $V$ is a (real) vector space endowed with the locally convex topology $\tau_{\mathcal{P}}$ induced by the given family $\...
fsp-b's user avatar
  • 1,064
0 votes
0 answers
21 views

For any (not necessarily continuous) linear form $f$ and any convex set $C$, $\overline C\subseteq f^{-1}(\overline{f(C)})$

Let $C$ be a convex subset of a l.s.c space $E$. If $f$ is a continuous linear form on $E$, then $\overline C\subseteq f^{-1}(\overline{f(C)})$, indeed $\overline C$ is the intersection of all closed ...
P. Quinton's user avatar
  • 6,076
0 votes
0 answers
34 views

Kernels of seminorms

Suppose $p_1,\cdots, p_n: X \to \mathbb{R}$ are seminorms defined on a vector space $X$, and $q$ is another non-zero seminorm such that $$\{0\}\neq \bigcap_{i=1}^n \text{Ker}(p_i)\subset \text{Ker}(q)....
user760's user avatar
  • 1,670
0 votes
0 answers
49 views

Continuity of an infinite matrix

Let $w$ be the space of sucessions with coefficients in $\mathbb{K}$, with $\mathbb{K}= \mathbb{R}$ or $\mathbb{C}$. For $x \in w$ ,let $p_n(x)=\max_{1 \leq i \leq n} |x_i|$ be a seminorm and consider ...
Contrad's user avatar
  • 47
1 vote
0 answers
95 views

$C(X)$ is a locally convex space - exercise 4.1.5. from Conway's Functional Analysis

The author defines locally convex sets as a TVS whose topology is defined by a family of seminorms $\mathcal{P}$ such that $\bigcap_{p \in \mathcal{P}} \{x\ |\ p(x) = 0\} = \{0\}$. A bit later he ...
the_dude's user avatar
  • 596
4 votes
1 answer
215 views

Why is a certain projective limit of weighted symmetric Fock space, namely $\bigcap\limits_{\tau \in T, p\ge 1 } \mathcal{F}(H_\tau,p)$, separable

I have a question regarding separability of a certain locally convex space. Let $H_{\tau}:=H^{\tau_1}(\mathbb{R}^n,\tau_2(x)dx)$ the weighted Sobolev Hilbert space with $\tau_1 \in \mathbb{N}, \tau_2(...
CoffeeArabica's user avatar
6 votes
1 answer
186 views

Convexity of a connected, compact, and locally convex set in $\mathbb{R}^n$

Is a compact, connected, and "locally convex" set in $\mathbb{R}^n$ convex? Here I mean a space $A$ locally convex as: For any point $x\in A$, there exists a neighborhood $U$ of $x$ s.t. $U$...
skt_zheng's user avatar
2 votes
1 answer
245 views

Understanding the locally convex topology induced by a family of seminorms intuitively

In our functional analysis course, we defined the locally convex topology induced by a family of seminorms as follows: Let $X$ be a vector space, $I \neq \emptyset$ an index set and $\{p_{\alpha}\}_{\...
ATW's user avatar
  • 677
2 votes
1 answer
33 views

Critical point for composite function implies $f'(x_{0}) = 0$ in locally convex spaces

Let $X, Y$ be locally convex spaces over $\mathbb{R}$ and $f: X \to Y$. Suppose $f$ is Fréchet differentiable at a point $x_{0} \in X$. Given an interval $0 \in I \subset \mathbb{R}$, we call a smooth ...
InMathweTrust's user avatar
0 votes
0 answers
49 views

Equality of Seminorms in vector bundles

In this notes Geometric Wave Equations by Stefan Waldmann at page 7 he has Let $E \longrightarrow M$ be a vector bundle of rank $N$. For a chart $(U, \psi)$ we consider a compact subset $K \subseteq ...
amilton moreira's user avatar
2 votes
1 answer
119 views

Regarding a non-standard definition of tempered distributions in Schuller's lecture

Is the below definition of tempered distributions correct? $\newcommand{\rr}{\mathbb{R}} \newcommand{\cc}{\mathbb{C}} \newcommand{\nn}{\mathbb{N}_0} \newcommand{\schwartz}{\mathcal{S}} \newcommand{\...
Apoorv Potnis's user avatar
2 votes
1 answer
61 views

Textbook on TVS that contains Theorem 32.2 about the completeness of the space of continuous linear maps

I have come across this thread about TVS. The OP took below screenshot of THEOREM 32.2. Could you please elaborate on the book from which the screenshot was taken?
Akira's user avatar
  • 17.6k
1 vote
0 answers
54 views

Existence of measure with given Fourier transform

Bogachev's "Gaussian measures" contains the following theorem: Theorem 3.3.1: Let $\mathcal{X}$ be a locally convex space and $G\subset \mathcal{X}^\ast$ a linear subspace separating the ...
George Gavrilopoulos's user avatar
1 vote
1 answer
84 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
0 votes
0 answers
46 views

Equivalent definitions of bounded sets in topological vector spaces

I want to show equivalence of following two definitions: Definition 1: A subset $U$ of a topological vector space is called bounded, if for every neighborhood of $0$ $V$, there is a scalar $s\in\...
user408858's user avatar
  • 3,130
5 votes
1 answer
131 views

Understanding equivalent definitions of locally convex spaces

As stated in the title, I am trying to make sense of the equivalent definitions of locally convex spaces. Especially, I am confused about the part, where I try to prove, that the topologies coincide. ...
user408858's user avatar
  • 3,130
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
0 votes
1 answer
31 views

$\varphi\mapsto x^\alpha \varphi$ is a continuous endomorphism in Schwartz space

I'm following some notes on functional analysis where it's shown the result presented on the title. The proofs in the notes is not clear to me. Before showing the proof I give some context. The ...
Davide Modesto's user avatar
0 votes
1 answer
55 views

$x_\beta \rightarrow x$ in a locally convex space if and only if $\rho_\alpha(x_\beta, x) \rightarrow 0$ for every seminorm $\rho_\alpha$

Let $X$ be a locally convex space with the family of seminorms $\{p_\alpha\}$. I am trying to get a feel for convergence of nets (or sequences) in these spaces aside from the general topological ...
CBBAM's user avatar
  • 6,295
0 votes
2 answers
63 views

Unit semiball of the supremum of seminorms is closed

I don't know how to prove the second part of this exercise (ex. 7.2 in F. Treves, "Topological vector spaces, distributions and kernels"). Let $\mathcal{P}$ be a family of continuous ...
david_sap's user avatar
  • 519
1 vote
0 answers
169 views

Confusion about locally convex (topological vector) spaces

The author in the book I am reading defines a locally convex space as follows. Definition: A topological vector space $(X,\tau)$ is called a locally convex space if there is an index set $A$ and a ...
user408858's user avatar
  • 3,130
2 votes
1 answer
78 views

Subbase for initial topology of quotient mappings (locally convex spaces)

Consider a family $\{p_i\}_{i\in I}$ of seminorms defined on $X$ and consider for each $i\in I$ the quotient mapping $q_i:X\rightarrow X/\ker(p_i)$. I know, that each quotient $X/\ker(p_i)$ is a ...
user408858's user avatar
  • 3,130
0 votes
1 answer
87 views

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 ...
CBBAM's user avatar
  • 6,295

1
2 3 4 5
11