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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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, ...
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1answer
15 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 ...
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2answers
36 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 ...
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49 views

Definition of the convolution with tempered distributions and Schwartz function

If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=\langle \tau_x \widetilde{\varphi} , u \rangle$, where we place ...
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0answers
24 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) ...
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2answers
31 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 ...
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9 views

Positive linear functional between cones

Let $W\subset V$ closed cones in $\mathbb{R}^n$. Is true that there exists a functional $f: \mathbb{R}^n\to \mathbb{R}$ such that $f$ is strictly positive in W but negative in some point of V? What if ...
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1answer
51 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 ...
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23 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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1answer
22 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 ...
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10 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 ...
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44 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
29 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?
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1answer
111 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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63 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 ...
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35 views

rconvex image under nonlinear function

Let $X\subset R^3$ be a compact and convex set, and let $f: X\rightarrow R^3$ be a nonlinear function, with $f\in C^k$. What are the tools to investigate if the image $K=f(X)$ is also convex, in the ...
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1answer
41 views

Open ball around a point in the hyperplane in a locally convex space

Let $f$ be a continuous linear functional in a locally convex space $X$ and let $z\in X$ so that $f(z)=\alpha$. Let $V_z$ be an open neighborhood around $z$. It is my intuition that $V_z$ should ...
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0answers
15 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 $X$?
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1answer
28 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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24 views

Is the subdifferential always convex and closed set?

Two properties of the subdifferential set are stated as follows: Given a function $f : \mathbb{R}^n → \mathbb{R}$, (i) the subdifferential set $\partial f(x)$ is always convex and closed, even if ...
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77 views

Continuity of Minkowski functional in locally convex topological vector space

Let $X$ be a locally convex topological vector space over $\mathbb{R}$ or $\mathbb{C}$ and let $p_C(x)=\inf (\lbrace t>0 \mid t^{-1}x \in C\rbrace)$ be the Minkowski functional for an arbitrary ...
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27 views

Get the global minimum with functions convex in a subset of the domain; numerical methods.

I have a $C^\infty$ function $f:\mathbb{R}^n\to \mathbb{R}$ that is positive and known to have a zero and a global minimum in an unknown point $x$. Furthermore, $f$ is convex in the set $$ S = ...
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0answers
28 views

Showing two topologies are equal

This is a homework problem from my real analysis class: I've been able to solve everything except to show that the two topologies are equal. I know the metric topology is the topology generated by ...
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1answer
83 views

Universal property of topology of uniform convergence

What kind of universal property does the strong dual topology on $X'$ have, for $X$ being a locally convex space. Is it possible to define $X'$ as the projective limit of the normed spaces ...
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20 views

Faces are hyperplanar

Let $P$ be a convex set with $0\in P^\circ$. Definition: We call a convex set $F$ a face of $P$ if $F\subsetneq\overline{P}$ with the property that for any $x,y\in\overline{P}$, if $(x,y)\cap F\neq ...
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29 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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80 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} ...
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43 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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1answer
18 views

Functionals on locally convex space of complex polynomials

Let $\mathcal{Z}=\{z_n\}_{n=1}^{\infty}$ be an infinite subset of complex numbers and $\mathfrak{P}:=\{\varphi_z \ : \ \varphi_z(p):=|p(z)|\}_{z\in\mathcal{Z}}$ a separating family of seminorms ...
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60 views

Example of a topological vector space which is not locally convex

I'm currently studying Functional Analysis and the professor gave an example for a TVS (which we have defined to be a vector-space $X$ in which addition $X \times X \rightarrow X, (x, y) \mapsto x + ...
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1answer
13 views

Semi-norm generating the standard locally convex topology on the space of locally finite Borel measures

In several articles available over the internet, it is written that: .. $M_{loc}(\Omega)$ is the space of locally finite Borel Measures on $\Omega$ with the standard locally convex topology ...
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1answer
48 views

Dual topology and Mackey–Arens theorem

I read only by wikipedia the Mackey–Arens theorem, that is: Given dual pair $(X, X')$ with $X$ a locally convex space and $X'$ its continuous dual, then $\mathcal{T}$ is a dual topology on $X$ if ...
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37 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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1answer
82 views

Equivalent definitions of locally convex topological vector space

This Wikipedia article gives two equivalent definitions of locally convex space (l.c.s). I don't see clearly the equivalence and I'd like to make it crystal clear. Definition 1 Let $(V,\tau)$ be a ...
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70 views

Continuity of seminorms

The following is from Wikipedia: A locally convex space is defined to be a vector space $V$ along with a family of seminorms $\{p_α\}_{α ∈ A}$ on $V$. A locally convex space carries a natural ...
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29 views

$f$ is concave, $g:\Bbb R \to \Bbb R$ is decreasing, prove that $g\circ f$ is convex

Please prove that if the function $f$ is concave and $g:\Bbb R \to \Bbb R$ is decreasing, then $g\circ f$ is convex.
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31 views

Why is $\{x-y: x,y\in K,\ d(x,y)\geq 1/n\}$ a compact set in a metrizable topological space with compact $K$?

Assume that $K$ is a compact convex set in a Hausdorff locally convex space, and $K$ is metrizable with the induced topology. Let $d$ be a metric defining the induced topology. Show that the set ...
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1answer
15 views

Show that $ F_n=\{x\in K:\textrm{there exist }\ y,z\in K\textrm{ such that }x=(y+z)/2\textrm{ and }d(x,y)\geq 1/n\}$ is a closed set

This is a follow-up question to a previous one. Assume that $K$ is a compact convex set in a Hausdorff locally convex space, and $K$ is metrizable with the induced topology. Then the set ...
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1answer
30 views

properties of non-extreme points

I'm reading a proof of the following lemma Assume that $K$ is a compact convex set in a Hausdorff locally convex space, and $K$ is metrizable with the induced topology. Then the set ...
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1answer
11 views

convex polytopes where every vertex pairwise shares a facet

for arbitrary dimension, what are the convex polytopes such that all vertices share a facet of some dimension, which is not the top facet (the entire polytope), with all other vertices? One example is ...
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1answer
134 views

Tempered distributions and convergence

It is known that the Schwartz class $\mathcal{S}(\mathbb{R}^n)$ is a Fréchet space and also that the space of test functions $\mathcal{D}(\mathbb{R}^n)$ is dense in $\mathcal{S}(\mathbb{R}^n)$. Let ...
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34 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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32 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 ...
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79 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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1answer
41 views

properties of a separable metrizable locally convex space

Let $X$ be a separable, metrizable locally convex space. Suppose $V$ is a neighborhood of $0$ and a barrel (closed, absolutely convex, and absorbing). Show that there exist points $y_n\in ...
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1answer
58 views

In the Krein-Milman theorem, can the weak closure of the convex hull be replaced by norm-closure?

I have a question on the following formulation of the Krein-Milman theorem: Consider a vector space $X$ equipped with the weak topology induced by a separating space $X^*$ of functionals on $X$. ...
3
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1answer
324 views

How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case?

I am reading a theorem about measurability of vector-valued functions in a note on functional analysis: Theorem 3.6.1. If $X$ is a separable, metrizable locally convex space, $(\Omega, \Sigma, ...
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1answer
213 views

Continuous inclusions in locally convex spaces

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C ...
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1answer
43 views

what is a cofinal sequence?

I understand that the subset $\Phi'$ of $\Phi$ is cofinal by looking at Wikipedia https://en.wikipedia.org/wiki/Cofinal_(mathematics) Would anybody explain what the cofinal sequence $(Y_n)$ means? ...
3
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1answer
67 views

How are the assumptions used in the proof of Bourbaki-Alaoglu Theorem?

This is a follow up question to a previous one. In the proof of the following theorem, where are the assumptions "Hausdorff" and "locally convex" used?