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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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33 views

Confusion: Tensor Norm on Lp spaces

In probability theory, I have often come across the identification $$ L^p_{\mathbb{P}}({\mathcal{F}})\otimes L^p_{\mathbb{P}}({\mathcal{F}}) \cong L^p_{\mathbb{P}\otimes \mathbb{P}}({\mathcal{F}\...
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15 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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1answer
60 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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16 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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1answer
31 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 ...
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27 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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45 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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43 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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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 ...
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26 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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49 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 $\...
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331 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 ...
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75 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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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 $\cap_{z\in N_{x}}T(z)\neq\emptyset$. ...
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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\...
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74 views

Approximation property for the quotient spaces of $C(T)$

As is known, for any compact space $T$ the Banach space $C(T)$ of all continuous functions on $T$ has the approximation property (see e.g. Albrecht Pietsch, Operator ideals). Is the same true for the (...
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1answer
44 views

Basis locally convex topological linear spaces

Let $X$ be a Hausdorff locally convex linear space, and denote by $\mathcal{N}_{0}$ the class of its (say, closed and absolutely convex) basis neighborhood of zero. Then, can we construct a sequence $...
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68 views

A convex function with a non-empty domain interior in a non-barreled locally convex space

Given a (Hausdorff separated) locally convex space $X$ what can we say about a proper convex function $f:X\to\mathbb{R}$ whose domain $\emptyset\neq D(f):=\{x\in X\mid f(x)<+\infty\}$ has a non-...
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78 views

About continuity on the space of measures.

Consider a separable, metrizable and locally compact space $\mathrm{X},$ that is not compact. Define $\mathrm{M}_{\mathbf{C}}(\mathrm{X})$ to be the set of complex measures on $\mathrm{X}$ and regard ...
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104 views

Over the separation of convex sets in a Banach space

I am working with Banach spaces and aim to prove the following property: If $A$ and $B$ are disjoint convex sets of a Banach space $X$ with $A$ open, then $A$ and $B$ can be separated, that is, ...
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40 views

What are weakest conditions on a vector space that allow defining dual (adjoint) operator?

I have an infinite dimensional topological vector space $V$ and its dual $V^*$ endowed with a sesquilinear form $$\phi:V^*\times V\to \mathbb{C}: (u,v)\to \phi(u,v) , \qquad u\in V^* , v\in V$$ I've ...
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81 views

Closed graph theorem for operator topologies - Do operator topologies yield Fréchet spaces?

Consider the SOT and WOT operator topologies on $B(H)$, the bounded operators on a hilbert space $H$. I'm interested in the properties of the topological vector spaces induces on $B(H)$. Are they ...
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29 views

Why is $\bigcup\limits_{|c| \leq 1} cU$ convex?

I have a question about proposition in this set of notes (https://www.math.ubc.ca/~cass/research/pdf/TVS.pdf) on topological vector spaces. We are in a complex vector space $V$. A subset $S$ is said ...
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77 views

Prove limit exists for an almost monotonic bounded function, involves convexity and square integrability

Let us consider the dynamics $\dot{x}(t)=-\gamma(t)y(t)\left(y(t)+m(t)\right)$ where $\gamma(t)\in [0,1]$ is a convex function that gaurantees the bound $\vert x(t)\vert \leq x_{\max}<\infty$ ...
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0answers
48 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. ...
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1answer
19 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 ...
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0answers
47 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 ...
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360 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}(\...
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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{\...
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211 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". ...
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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-...
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61 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 ...
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1answer
259 views

Goldstine theorem and weak topology on $E$ induced by weak* topology on $E''$

In the book where I'm studying there this exercise: "(Goldstine's theorem). Recall that $E \subset E''$, or, more precisely, $J_E(E)$ is a subspace of $E''$ where $J_E :E \longrightarrow E''$ is ...
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1answer
54 views

Faithfullness of the Minkowski functional

Let $X$ be a locally convex topological vector space. I need to show that the Minkowski functional $p_C$ for $C$ a convex open neighborhood of $0$ coming from the local base of convex sets, is ...
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0answers
43 views

Conflict with definition of “face”

I am given this definition of face from Convexity: An analytic viewpoint: Definition: A face of a convex set $P$ is a set $F\subsetneq \overline{P}$ such that for every $x,y\in\overline{P}$ and for ...
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56 views

Existence of linear continuous functional on locally convex TVS

Let $c>1$ and $X$ be a locally convex TVS. Take $a,b\in X$ and a closed subspace $Y$ of $X$ such that $Y\cap \{(1-t)a+bt \ | \ t\in\mathbb{R}\}=\emptyset$. How to prove that there exist $f\in X'$ ...
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2answers
67 views

A subset $K$ of $L^1$ such that is convex, absorbent and balanced, but not neighborhood of $0$.

It is well-known that, if $(\Omega,\mathcal{F},P)$ is an atomless probability space, then $L^1$ is barreled, in the sense that every subset $K$ which is closed, convex, absorbent and balanced is a ...
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0answers
207 views

Show that $f$(sum of log terms) is convex with Jensen's inequality..

We have the equation \begin{equation} f(\mathbf{x})=\mathbf{c}^T\mathbf{x}-\sum_{i=1}^m\log{(b_i-\mathbf{a}_i^T\mathbf{x})} ,\;\;\; \mathbf{x} \in \mathbb{R}^n \text{ and } m>n \end{equation} ...
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0answers
136 views

Local boundedness of monotone operators in general spaces

A classical result states that: If $X$ is a Banach space then every multi-valued monotone operator $T:X\to 2^{X^*}$ is locally bounded on $\operatorname*{int}D(T)$ (the interior of its domain). I ...
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0answers
229 views

Arzelà–Ascoli theorem for operators

If you have a net of continuous linear operators between reasonable spaces (complete, at least), does there exist Arzelà–Ascoli-like theorems giving convergent subnets? I believe that it should be ...
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0answers
360 views

Half space representation of a convex polytope

We know that the half space representation of a polytope is given by: $Ax<b$. Consider a convex polytope in $\mathbb{R}^3_+$ with vertices given by the following set of points: $\left\{(0,0,0),(1,0,...
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0answers
210 views

Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$, . Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
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0answers
61 views

The element presentation of convex hull of the union of compact sets

I want to show that the convex hull of the union of compact convex sets $k_{1}$ and $k_{2}$ in a locally convex topology linear space, consist of the points of the form $$ay_1+(1-a)y_2,\ \ \ \ \ y_1\...
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0answers
68 views

Continuity of the dual product reloaded

Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $(x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R}$ is strongly$\times$strongly continuous in $X\times X^*$. That does not ...
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0answers
172 views

When is the normal cone to a closed convex set in a locally convex set maximal monotone?

Let $X$ be a locally convex set with the following property: (P) $\forall C\subset X$ closed convex, the normal cone $N_C$ is maximal monotone as a multi-valued operator from $X$ to its topological ...
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1answer
117 views

Existence of a Frechet topology on the dual of a barreled space

I have a Hausdorff separated locally convex barreled space $(X,\tau)$ with topological dual $X^*$. My questions are: $Q_1$ Is there a topology $\tau^*$ on $X^*$ that is finer than the weak-star ...
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1answer
91 views

Proving a point is a local minimum

I have a rather basic question. I have a function $f:R \rightarrow R$, and I want to show a point, $x^*$, is local minimum, i.e., $f(x^*+\delta) \geq f(x^*), \ \delta \to 0$. I can show that: $f(x^* ...
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0answers
128 views

Direct sum decomposition of vector spaces and their tensor powers

Let $V$ be a locally convex vector space and let $U$ be a finite-dimensional subspace of $V$. The Hahn-Banach theorem guarantees that there exists a closed subspace $W$ of $V$ such that $$V=U\oplus W.$...
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0answers
38 views

Differential calculus on locally convex spaces

For real finite dimensional vector spaces $V,W,Z$, i know that a map $f:V \times W \to Z$ is smooth if the maps $f(v,.)$ and $f(.,w)$ are for every $v\in V, w \in W$. Does the same thing hold for (...
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0answers
416 views

The closed graph theorem for Banach spaces isn't true. True?

I'm reading through some functional analysis lecture notes and there the closed graph theorem was stated in the following form: Let $X$ be a Baire locally convex space and $Y$ a Frechet space. If ...