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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Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$

We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset ...
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67 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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39 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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16 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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37 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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9 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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31 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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36 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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65 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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59 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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28 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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27 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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54 views

A maximal monotone operator (and not a subdifferential) with a non-convex domain

I am looking for a maximal monotone operator $T:X\to 2^{X^*}$ that has a non-convex domain $D(T)=\{x\in X\mid T(x)\neq\emptyset\}$ and $T$ is not cyclically monotone, that is, $T$ is not a convex ...
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12 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
27 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
9 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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72 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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30 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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31 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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72 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
37 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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45 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 $X$. ...
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310 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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164 views

Inclusions continuous space of test functions

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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31 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? ...
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51 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?
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97 views

Closedness in the proof of the Alaoglu Theorem

I'm reading a proof the Bourbaki-Alaoglu Theorem: Could someone explain how the closedness of $\Phi(V^\circ)$ (namely, $\Phi(V^\circ)$ contains all of its limit points) is done in the proof? I ...
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44 views

neighborhood base for the Mackey topology

I'm reading the proof of a theorem due to Mackey in a note of functional analysis: I don't see why the first sentence is clear. By the definition of neighborhood base and locally convex topology, ...
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31 views

weak topology and dual pairs

I'm reading a note on functional analysis and the following statement is given without a proof: Let $(X,Y,\langle,\rangle)$ be a dual pair and $\tau$ a topology on $X$. Let $\sigma(X,Y)$ be the ...
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30 views

In a proof of the representation of linear functionals of topological vector space

I'm reading the proof of the following theorem in a note on functional analysis: Here $p_F$ is defined as $p_F(x)=\max_{y\in F}|\langle x,y\rangle|$. Could anyone show me why the underscored ...
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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 ...
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27 views

rigorously show the given function is either convex or non convex.

Consider $f:\mathbb{R}^{3}\rightarrow\mathbb{R}$ such that $(x,y)\mapsto \ln(e^{x}+e^y+e^z)$. Determine whether or not $f$ is convex and rigorously justify your answer. I am thinking this function is ...
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Expressing continuity in terms of seminorms

Let $X$ and $Y$ be locally-convex topological vector spaces, with topologies given by families of seminorms $(p_i) _{i \in I}$ and $(q_j) _{j \in J}$, respectively. If $L : X \to Y$ is a continuous ...
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66 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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1answer
97 views

Can a smooth convex functions be non-differentiable?

Consider the definition of the $\beta$-smoothness (for some constant $\beta$): $$ \|\left. \nabla f \right|_{ y } - \left. \nabla f \right|_{ x } \| \leq \beta \| x - y \| $$ And convexity: $$ ...
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Reversing the search for a convex hull

In the wikipedia article on convex hulls, there is an image showing a rubber band shrinking down to form a polygon around a set of points in a plane. I have a set of data with a region with no points ...
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105 views

Example of Topological Vector Space

Is there a topological vector space such that, for every $x\in X$, there is a proper neighbourhood $V$ of $x$ in $X$ which is convex, but the whole space is not locally convex (i.e. $X$ has a local ...
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17 views

Unit vectors in locally convex spaces

Are there unit vectors in locally convex spaces. If yes, how can vectors be normalised in locally convex spaces?
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79 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 ...
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31 views

Where can I find the nowhere subdifferentiable example of rockafellar?

I'm told that Rockafellar gave an example of a real extended function defined on a locally convex space, whose subdifferential is empty at each point of its domain. The function is proper, lower ...
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43 views

Dual of continuous functions in various topologies

Let $S$ be compact and Hausdorff and $C(S)$ be its space of continuous complex functions. When $C(S)$ is endowed with the $\sup$ norm, its dual is well known. Since this topology is too strong for my ...
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65 views

Topology on compactly supported smooth functions

I'm confused by a set of lecture notes I'm reading and would like help in understanding what's going on. First, there is the following nice theorem. Theorem. The topology of a locally convex space is ...
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Suppose $P = \text{conv}(\text{ext}(P))$. Is it possible to find $S \subset P$ of cardinality $< \text{ext}(P)$ s.t $\text{conv}(S) =P$?

Suppose that $P$ is a polyhedron and that $P = \text{conv}(\text{ext}(P))$. Suppose further that $ \text{ext}(P)$ has cardinality $r$. How do I see that I cannot find a set $S \subset P$ of ...
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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 ...
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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 ...
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73 views

Characterize polytopes resulting from cutting a convex polytope by a hyperplane

We have a convex polytope $P$ for which we know its set of vertices. Using this set we characterize the H-representation of $P$ as: $\mathbf{A}\mathbf{x} < \mathbf{b}$. If a hyperplane defined by ...
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82 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: ...
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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 ...
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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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66 views

Approximate model of a convex/concave surface

I have a set of measurements in 3d that yields a concave surface of a function $f(x,y)$ that I don't know its expression. I am thinking to approximate the function to a model where any point from the ...