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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).

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1answer
69 views

convexity of a relatively open subset of a compact set

I'm struggling with the following problem: it seems to be true but I'm not able to prove it! Let $C$ be a compact convex subset of a locally convex metric vector space and $\hat{C}$ be a relatively ...
1
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1answer
53 views

About the locally convex topology

I know that if a locally convex space Hausdorff $(X,S)$ is first numerable then for the $\hat{0}\in X$ exists a countable local base $\{V_n, n \in \mathbb{N}\}$ and to each $V_n$ corresponds a ...
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0answers
30 views

Properties of topological vector spaces

I'd like to understand better the significance of certain properties of topological vector spaces. It would be great if someone could give me an intuitive picture for what makes them "special", and/or ...
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0answers
24 views

Product of open maps open? (not the cartesian)

Let $A$ be a locally convex algebra, or even just a topological algebra, and let $U_1,U_2\in A$ be open, is the product $$ U_1\cdot U_2=\left\{ a\cdot b\mid a\in U_{ 1} ,b\in U_{ 2} \right\} $$ ...
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0answers
12 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 ...
0
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1answer
42 views

This convex hull is balanced?

Let be (X,S) a locally convex space, and $B \subset X$ a nonempty sequentially closed bounded and convex set such that $\hat{0} \notin clB$,(the closure of B). Define the set T:=s-clco$\{B \cup-B \}$ ...
4
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2answers
70 views

Convergence on locally convex spaces

I'm new on the locally convex spaces. I know that if $X$ is a vector space and $S$ an irreducible set of seminorms defined in $X$, $(X,S)$ is a locally convex vector space. The first question is, how ...
0
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1answer
45 views

Convex function attains maximum at extreme point

So I am working on the proof that a convex continuous function $f$ on a convex and compact set $X$ attains its maximum at an extreme point of $X\subset\Omega$ where $\Omega$ is a locally convex space ...
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1answer
29 views

Two notions of boundedness in metrizable topological vector space.

In a metrizable topological vector space $X $ with the metric $d $, a subset A is said to be bounded if it can be absorbed by any neighbourhood of $0$ and a subset A is said to be d-bounded if its ...
0
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1answer
36 views

A theorem about convex function

Assume that function $h(x)=f(ax+b)$ is a convex function. What can we say about the convexity of function $f(x)$? My notes: By taking the second derivative from both sides of eqaution $h(x)=f(ax+b)$ ...
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0answers
43 views

Rudin's functional analysis Theorem 3.18, second part.

Just a follow up to the following two questions: Rudin's functional analysis 3.18, every originally bounded subset of a locally convex space is weakly bounded. Theorem 3.18, Rudin's ...
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1answer
30 views

Rudin's functional analysis 3.18, every originally bounded subset of a locally convex space is weakly bounded.

Reading through the proof of the following In a locally convex space $X$, every weakly bounded set is originally bounded, and viceversa The trivial part of the proof. Since every weak ...
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0answers
16 views

About Locally Convex Frèchet Space

I need to proof the following statements, having $ \quad X = \{x_n : \mathbb{N} \rightarrow \mathbb{C}\}$ and $p_j = max_{k \le j}|x(k)|$ 1)$p_j$ is a countable family of seminorms which induces ...
0
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1answer
31 views

Locally Convex tvs closure of $\{0\}$

Let $E$ be a topological vector space locally convex, defined by the family of seminorms $\mathcal{F}=(p_j)_{j\in J}$. I can't prove that $\underset{j\in J}\bigcap Ker(p_j)=\overline{\{0\}}$
0
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1answer
32 views

On the completeness of topologically isomorphic spaces

Let $(E_1,\tau_1)$ be a locally convex space and let $(E_2,\tau_2)$ be a complete locally convex space. Suppose that $T:(E_1,\tau_1) \longrightarrow (E_2,\tau_2)$ is a topological isomorphism (that is,...
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0answers
17 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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0answers
40 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. ...
2
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1answer
24 views

Is the space of bounded continuous functions on the reals with the topology of uniform convergence on compact sets fully barrelled?

Let us consider the space $X=C_b(\mathbb{R},\mathbb{R})$ endowed with the following topology: $f_n\rightarrow f$ iff $f_n\rightarrow f$ uniformly on compact subsets of $\mathbb{R}$. This space is a ...
0
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1answer
73 views

What is the (strong) dual of a projective/inductive limit?

For a family of locally convex TVS $(E_\alpha)_{\alpha \in A}$ it is a well known fact that the dual of the product is the sum of the individual duals and vice versa, i.e. $$ \left(\prod_{\alpha \in A}...
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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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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 ...
2
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3answers
56 views

Is a linear functional on a locally convex space $X$ that continuous on a dense subset of X continuous?

Let $X$ be a locally convex space and let $M$ be a dense subset of $X$. Let $f$ be a linear functional on $X$ such that $f$ is continuous on $M$. Then is $f$ continuous on $X$? Thank you in advance!
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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
46 views

How to extend $f\in M^*$ to $\overline{M}$? [closed]

Let $X$ be a locally convex space, let $M$ be a linear subspace of $X$, and let $f\in M^*$. If $M$ is closed, then $M=\overline{M}$. And we don't have to extend $f$. So how to extend $f$ to $\...
2
votes
1answer
107 views

One assumption in the proof of one result of Hahn-Banach theorem.

Theorem: Let $X$ be a locally convex space, let $M$ be a linear subspace of $X$, and let $f\in M^*$. Then there exists $h\in X^*$ such that $h(x)=f(x)$ for all $x\in M$. I saw one sentence in the ...
2
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2answers
42 views

Joint continuity of bilinear pairing (with unusual topology)

Let $V$ be a complex vector space, which we endow with the finest linear topology. Then continuous dual $V'$ coincides with the algebraic dual $V^*$. We choose the weak-star topology $\sigma(V^*,V)$ ...
3
votes
1answer
96 views

How to show $X := \bigcup_{n \in \mathbb{N}}X_n$ is a locally convex space.

My textbook on functional analysis says as follows.(The book is written in Japanese, ISBN: 978-4-946552-18-2) Let $\{X_n\}$ be a sequence of locally convex spaces over $K (= \mathbb{R}$ or $\mathbb{C}...
2
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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 $\...
1
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1answer
51 views

A particular TVS

I am looking for a topological vector space $X$ satisfying in the following properties: (1) Cardinal number of $X$ is at most of continuum. (2) $X$ is not a hereditary Lindelof space. (3) $X$ ...
0
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1answer
27 views

Reference request: Locally convex space is Hausdorff if and only if …

I was given the following theorem by my professor: Let $A$ be a locally convex space. Then $A$ is Hausdorff if and only if, for every seminorm $p$, we have that $p(x)=0$ implies that $x=0$. If I ...
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2answers
34 views

Is a countable product locally convex?

Let $X$ be a countable set. Consider the space $\mathbb{R}^X$ of real-valued functions on $X$ equipped with the product topology. Is $\mathbb{R}^X$ locally convex? If not, is the space of real-...
1
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1answer
58 views

A linear subspace $Y$ is dense iff there is no trivial funcitonal vanishing on $Y$

So I was reading Conway's book "A course in functional analysis" and stumbled upon the following corollary of the Hahn-Banach separation theorem: If $X$ is a locally convex space and $Y$ is a ...
0
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1answer
30 views

Locally convex vector space: Unified polars of zero neighbourhoods are the dual space

This is a statement which I found without proof and maybe it's obvious, but I can't understand why it should be true: Let $X$ be a locally convex vector space, $\mathcal{W}$ a neighbourhood basis ...
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0answers
233 views

a list of known convex functions?

I am working on the development of novel optimization algorithm where it DOES NOT suffer from the non-differentiability (cf. subgradient), which also works well in differentiable class of the ...
0
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1answer
294 views

Is the normal cone of a polyhedron a set of points or a set of vectors?

I looked for the normal cone to a polyhedron, and I found this definition from https://sites.math.washington.edu/~rtr/papers/rtr169-VarAnalysis-RockWets.pdf: But this is confusing to me. it seems to ...
0
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1answer
43 views

Existence of continuous norm on C(X)

Let $X$ be a metrizable topological space, and $C(X)$ the space of continuous functions. Is there a continuous norm (as function to $\mathbb{R}$) on $C(X)$? The topology is given by the family of ...
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2answers
89 views

Is infimum distributive for norms?

I am trying to prove that the point to set distance when the set $S$ is a convex cone, is sub-additive: $K$ is a cone, and $x,y\in K$ $dist(x+y|K) \le dist(x|K) + dist(y|K)$ where $dist(y|K)=\inf_{...
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1answer
28 views

Why is $B$ closed and bounded in $A$?

Let $A$ be a locally convex algebra. By $\mathcal{B}_1$, we denote the collection of all subsets $B$ of $A$ such that: $B$ is absolutely convex and $B^2\subseteq B$ $B$ is closed and bounded ...
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0answers
68 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, \...
2
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1answer
82 views

Locally convex inductive limit topology versus cofinal topology

Let $V$ be a complex vector space and let $\{0\} \subset V_1 \subset V_2 \subset \dots \subset V$ be an increasing sequences of subspaces of $V$, whose union is $V$. Suppose that each $V_n$ is a ...
1
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1answer
47 views

tensor product of locally convex spaces

Let $X$ and $Y$ be locally convex vector spaces, over $\mathbb{C}$, equipped with directed families of seminorms $\mathcal{P}_X$ and $\mathcal{P}_Y$ respectively inducing the topologies. For each ...
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1answer
101 views

The space of measurable functions is Frechet?

Take a bounded set $S\subseteq \mathbb R^n$ with non-zero measure, and $M_S$ the set of measurable complex functions over $S$. We know that the convergence in measure is metrizable and complete. ...
0
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1answer
16 views

Why is $e \in C$ (a commutative subalgebra of $A$)?

I am reading the following proposition: Here $e$ represents the identity element of $A$ and $\sigma_C(x)$ and $\sigma_A(x)$ denote the spectrum of an element $x$ in $C$ and $A$ respectively. ...
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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 $\cap_{z\in N_{x}}T(z)\neq\emptyset$. ...
4
votes
1answer
86 views

It is always possible to define a topology in a vector space endow with a semi-norm?

If $(X,\|\cdot\|)$ is a semi-normed vector space. It is always possible to define a topology on $X$? If it is true What is the definition of a closed subspace of $X$ with respect to $\|\cdot\|$? ...
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2answers
54 views

Topological isomorphism between $C^{\infty}(\mathbb{R}) = \lim_{\leftarrow}{C^{k}([-k, k])}$

Let $X = C^{\infty}(\mathbb{R})$ be the space of smooth functions on a real line considered as a locally convex topological vector space, endowed with the family of seminorms $$p_{m}(\varphi) = \max \...
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1answer
73 views

Non-Hausdorff topology on the germs of holomorphic functions

Let $\mathcal{O}_{z}$ be the space of germs of holomorphic functions at $z$, defined as a direct limit of a system $$\mathcal{O}_{z} = \lim_{\rightarrow}{\mathcal{O}(U_{\frac{1}{n}}(z))}$$ where $U_{\...
0
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1answer
171 views

The finest locally convex topology is not metrizable

Let $E$ be a vector space. Recall that the inductive topology in $E$ taken w.r.t to the family of spaces $\{ E_{\alpha} \}$ and mappings $g_{\alpha}: E_{\alpha} \rightarrow E$ is the finest topology ...
2
votes
2answers
57 views

Locally convex space with a *algebra structure

Consider a $*$-algebra $A$ (over the complex numbers $\mathbb{C}$), which is also a locally convex space, say by the separated family of norms $\{ n_i\}_{ i \in I }$. Assume that the $*$ automorphism ...
3
votes
1answer
136 views

Characterization of extreme points of $B_{C_0(X)^*}$

Let $X$ be a compact Hausdorff space and $C(X)$ the Banach space of continuous $\mathbb K$-valued functions equipped with the supremum norm. We denote the dual space of $C(X)$ by $C(X)^*$. A well-...