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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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254 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 $\...
5
votes
0answers
215 views

Isomorphism between $C^\infty_0(B_1)$ and $\mathscr{S}(\mathbb{R}^n)$

Background: Related question I am trying to prove, that the countably-normed spaces $C^\infty_0(B_1)$ on the open unit ball (i.e. function and all derivatives vanish at the boundary) in $\mathbb{R}^n$...
4
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0answers
218 views

A weakly closed set in $X$ that remains weakly-star closed in $X^{**}$

I have $X$ a (non-reflexive) Banach space and $B\subset X$ a weakly closed convex subset. I wonder under what additional conditions (other than weak compactness) $B$ remains weakly-star closed in $X^{...
4
votes
0answers
112 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 ...
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 ...
3
votes
0answers
217 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 $p$. ...
3
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0answers
99 views

Show that the “folium” is norm closed

A C*-algebra $\mathfrak{A}$ is a Banach algebra with an involution operation $* : \left\lbrace\begin{aligned} \mathfrak{A} &\longrightarrow \mathfrak{A} \\ a &\longmapsto a^* \end{aligned} \...
3
votes
0answers
70 views

In a proof of “weakly measurable implies measurable”

This is a follow-up question to the following ones: How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case? properties ...
3
votes
0answers
163 views

Are not all neighborhoods of $0$ in a locally convex space absorbent?

A locally convex space (LCS) can be defined as a topological vector space (i.e. scalar product and sum are continuous) whose topology is generated by translation of a family of balanced and absorbent ...
3
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0answers
128 views

Continuity of a function in a locally convex topological space

I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and ...
3
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0answers
89 views

Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
3
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0answers
106 views

Motivation for the notion of locally convex topological vector space

Is the only motivation for the notion of locally convex topological vector space that the local bases have some nice property i.e. convex, balanced, absorbing ?
3
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0answers
99 views

Strong Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
3
votes
0answers
117 views

Understanding bornologic spaces

The definition as it appears in Yosida's book: A locally convex space $X$ is called bornologic if it satisfies the condition: If a balanced convex set $M$ of $X$ absorbs every bounded set of $X$, ...
2
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0answers
35 views

Epsilon tensor product of locally convex spaces

I want to understand the definition of the $\varepsilon$-tensor product of two locally convex vector spaces. (Mainly as a hobby.) Let $X,Y$ be locally convex vector spaces and let $B(X^{\ast},Y^{\...
2
votes
0answers
37 views

Continuity and sequential continuity of a linear functional

Let $E = C_c^0(\mathbb{R}^n;\mathbb{R}^m)$ be the space of compactly supported continuous functions on $\mathbb{R}^n$ with values on $\mathbb{R}^m$. There is a natural norm on this space: given $\...
2
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0answers
36 views

Minimization of regularized functional

Background: Let $x,y$ be points in $\mathbb{R}^D$. Let $C$ be functionals from $\mathbb{R}^D$ to $\mathbb{R}$, which are strictly convex and convex respectively. Moreover, assume that $F$ is ...
2
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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 ...
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 $\...
2
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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....
2
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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=...
2
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0answers
48 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
0answers
237 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 ...
2
votes
0answers
147 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) ...
2
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0answers
22 views

Existence of Banach space in which nuclear space embeds densely

If $N$ is a nuclear space, does there exist a Banach space $X$, s.t. $N$ embeds densely in $X$?
2
votes
0answers
180 views

Are Lusin and Souslin spaces sequential or even Fréchet-Urysohn?

First some definitions: A Polish space is a separable and completely metrizable topological space. A Hausdorff space is Lusin if it is the image of a Polish space under a bijective continuous map. A ...
2
votes
0answers
495 views

Prove that the set of extreme points of a compact convex set is not empty.

The Krein–Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points. Is ...
2
votes
0answers
99 views

Relation between the number of halfspaces and the number of vertices of a convex polytope

Suppose we have an $n$-dimensional convex polytope $\mathbf{P}$ represented by an intersection of half-spaces as the following: \begin{equation} \mathbf{P} = \{ x \in \mathbb{R}^n \mid x \in \bigcap_{...
2
votes
0answers
157 views

Show that continuous linear maps on the space of test functions take $C_K^\infty(\Omega)$ into some $C_{K_N}^\infty(\Omega)$

Let $\Omega$ be a nonempty open subset of $\mathbb{R}^n$, and let $\cup_{n=1}^\infty K_n = \Omega$ be an exhaustion of $\Omega$ by compact sets. Let $\mathcal{D}(\Omega) = \mathcal{D}$ be the standard ...
2
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0answers
28 views

On the Space $C([a,b],X)$, where $X$ is LCS

Definition. A family $P$ of semi-norms on a vector space $X$ is called directed if for any $p_1,p_2\in P$ there exist $p\in P$ and $C>0$ such that $p_1(x)+p_2(x)\leq Cp(x)$ for all $x\in X$. Let $...
2
votes
0answers
335 views

If a line goes through the boundary of a convex set, does that line intersect with exactly two boundary points of the convex region?

I have found the following theorem that is often cited from the text Convex Figures by Yaglom and Boltyanskii: A bounded figure in $\mathbb{R}^2$ is convex iff every straight line passing through an ...
2
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0answers
70 views

Direct limits of locally convex spaces and embeddings

I was thinking about whether this positive result would hold in the category of locally convex spaces also... Here is what I got so far: The direct limit of a locally convex system consists of the ...
2
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0answers
55 views

Topological modules with enough continuous linear functionals.

Context: I'm trying to find out which topological (unital) modules are "good enough" for generalizing results from real or complex functional analysis. For example, I say that a module, in order to be ...
2
votes
0answers
1k views

Convex hull of bounded set is bounded

I want to show that in a locally convex topological vector space $X$, the convex hull of a bounded set is bounded. Apparantly this does not hold if $X$ is not locally convex. So the fact that that ...
2
votes
0answers
116 views

Does $\mathsf{Man}$ possess countable products?

In our lecture we defined a category $\mathcal C$ to have arbitrary products, iff every diagram $A:\mathcal J \to \mathcal C$ with $\text{Morph}(\mathcal J)=\{\text{id}_j\}_{j \in Ob(\mathcal J)}$ has ...
2
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0answers
208 views

closed subspaces of locally convex inductive limits

Let $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ be an inductive sequence of LCTV spaces. A locally convex inductive limit of $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ is it's ...
2
votes
0answers
92 views

Is a single nontrivial convex set in a topological vector space enough to make it locally convex?

This is sort of a definition question. While a tvs is locally bounded if it contains a bounded neighborhood of the origin, a tvs is called locally convex if it contains a fundamental system of ...
2
votes
0answers
119 views

Norm inequalities in a reflexive space

I am reading an article about reflexive spaces, with a specific example. The article mentions inequalities that I haven't been able to get around to. Here's the setup. The space $X = (\prod_n \...
2
votes
0answers
89 views

maps from a convex set to itself

Suppose $S\subset \mathbb{R}^n$ be a closed convex set under Euclidean topology (but not necessarily bounded, example a closed cone). Let $\mathcal{E}(S)=\{L:\mathbb{R}^n\rightarrow \mathbb{R}^n\text{ ...
2
votes
0answers
178 views

About continuous linear functional on the topology generated by linear functionals

I am self-studying P. Lax's functional analysis book. Here is an exercise in p123. it is supposed to be very easy, but I really couldn't see it. Could you anyone help me out? Thanks. Let $\{ l_\...
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vote
0answers
9 views

Projective tensor product on projective LCM is exact

I am reading the book "The Homology of Banach and Topological Algebras" by A.Y. Helemskii and couldn't understand one lemma on page 204 about complex splitting. I understood how to prove that complex ...
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0answers
13 views

Can I construct a monoidal category on locally convex modules over algebras with approximate identiy

I am faced with the following problem: Let $A$ be a complete locally convex algebra with a uniform approximation of identity, that is a net $e_\lambda$, such that $p(e_\lambda a-a)\rightarrow 0$ for ...
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0answers
19 views

Characterization of convex dual-like functions

Let $f$ be a proper, convex, lsc functional from, a locally-convex topological vector space, $X$ to $\mathbb{R}$. Then by Fenchel-Monreau Theorem, $$ f(x) =\sup \left \{ \left. \left\langle x^{\star} ...
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vote
0answers
41 views

Does Grothendieck's theorem hold for Bounded borel functions?

In the case of continuous functions on a compact Hausdorff space, we have that any bounded set is pointwise compact if and only if it is weakly compact and the two topologies coincide with this set. ...
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0answers
42 views

Is the extension in this answer well defined and linear?

I ask this question One assumption in the proof of one result of Hahn-Banach theorem. before. But I have trouble understanding the answer. I am not quite sure the extension in the answer is well ...
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vote
0answers
23 views

Is this extension continuous on $X$?

Let $X$ be a locally convex space. Let $M$ be a dense subspace of $X$ and let $f\in M^*$. And I am trying to show there exists $g\in X^*$ such that $g|_M=f$. My attempts are: Let $x\in X$. Then ...
1
vote
0answers
24 views

Try to find a seminorm $p$ on a locally convex space st $|f|\leq p$.

Let $X$ be a locally convex space. Let $M$ be a linear subspace of a locally convex space $X$. Let $f\in M^*$. Then can we find a seminorm $p$ on $X$ such that $|f(x)|\leq p(x)$ for all $x\in M$? ...
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0answers
70 views

A problem with Theorem 6.4 in Rudin's Functional Analysis

I feel a bit uneasy about the proof of the following Theorem in Rudin's Functional Analysis, 2nd edition, p. 152-153. It says that for the space $\mathcal{D}(\Omega)$ and a certain systems $ \beta, \...
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0answers
26 views

How do I prove the Local intersection property in the example(Economics)

$T$ is said to have local intersection property if for each $x\in X$ with $T(x)\neq\emptyset$, there exists an open neighborhood $N_{x}$ of $x$ such that $\cap_{z\in N_{x}}T(z)\neq\emptyset$. ...
1
vote
0answers
34 views

locally convex vector space, functional analysis, seperation theorem

Let $(E,\tau)$ be a locally convex $\mathbb{K}$-vector-space and $A\subseteq E$ with $0\in A$. The following statements are equivalent: (1) $A$ is closed and convex (2) It exists a subset $B\...