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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22 views

A Convex Function Inequality Problem

I would be grateful if someone can provide a proof of the following problem: Let $f(x)$ be convex function. Let $\lambda_1+\lambda_2+\lambda_3=1$, where $0\leq \lambda_k\leq 1$ for $k=1,2,3$. Let $\...
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25 views

Convexity of min(max()) operation over a convex set and a hypercube

Suppose we have a hypercube $[0, 1]^n$, a convex set $V$, and another set $A = min\{max\{V, \mathbf{0}\}, \mathbf{1}\}$, what should be the convexity of $A$? I've found out that $A$ is not a convex ...
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15 views

Confusion: Tensor norm between LCSs

Recently I have read that one may define a "tensor norm" on the projective tensor product $X\otimes_{\pi} Y$ between two locally convex spaces. If $X$ and $Y$ are not Banach spaces, how is this ...
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28 views

Hahn-Banach separation with point on boundary of a convex set

I want a separation result as follows; finite dimensions might not matter but is the case of present interest. Suppose $C\subset \mathbb R^n$ is convex (not necessarily open or closed) and $0\in \...
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28 views

Strongly Minkowski equivalence

Assume that $(X, \{ p_i \}_{i \in I})$ is a locally convex space. $A,B \subset X$ are said to be strongly Minkowski separated iff there exists $j \in I$ and $z \in X$ such that one of the following ...
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23 views

Prove that a function is sequentially lower semicontinuous

Let be $(X, \{ p_i \}_{i \in I} )$ a locally convex space, $M_0\subset X$ a bounded and nonempty set and $f = l + I_{M_0}$ where l is a continuous function and \begin{equation*} I_{M_0}(x)= \begin{...
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5 views

Generalization to $n$ dimensions of partition into convex subsets of a function’s domain?

Let $f:\mathbb R\to\mathbb R$ be a smooth and bounded function with a finite number of local minima. Then we can partition $\mathbb R$ into a finite number of sets $\{ A_i\}$, such that the ...
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18 views

direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps $$ 0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0. $$ We can take inductive limit (...
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29 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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39 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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53 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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17 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 ...
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38 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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127 views

Closed and convex subset of a locally convex space

For the following problem concerning functional analysis I cannot seem to find a solution. It states as follows. Let $(E, \tau)$ be a locally convex vector space and $S \subseteq E$ with $S \ni 0$. ...
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74 views

how to prove the following equation is convex funcion?

prove that $$ f(x,t)=\frac{\Vert x\Vert _p^p} {t^{p-1}} $$ is convex on $$ \{ (x,t)|x\in R^n ,t\ge 0 \} $$
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95 views

Is this true? A strict monotonic function F satisfying Jensen equation with F(0)=0, only expresses dyadic rationals multiples?

. Is is really true that a function satisfying jensen equation F(x/2+y/2)=F(x)/2+F(y)/2, F(0)=0, and F strictly monotonic, where F is a function from F:[0,1] to [0,1] can only express dyadic fraction ...
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52 views

$f|_{(a, b)}$ convex. Is $f$ convex on $[a, b]$? [closed]

Suppose $f : [a, b] \rightarrow R$ is continuous on $[a, b]$ and convex on the open interval $(a, b).$ Show that $f$ is convex on the closed interval $[a, b].$
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29 views

Showing $\forall q\in\mathcal{P}_Y$ there exists $ p\in\mathcal{P}_X:q\circ A\le p\implies A:X\to Y$ continuous

Let $(X,\mathcal{P}_X)$ and $(Y,\mathcal{P}_Y)$ be locally convex topological vector spaces with topologies induced by the families of continuous seminorms $\mathcal{P}_X$ and $\mathcal{P}_Y$ ...
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117 views

Application of the separation theorem of convex sets

Let $X$ be a locally convex topological vector space over $\mathbb{R}$, $a,b\in X$, and let $Y$ be a closed, subspace of $X$, such that: $$Y\cap\left\{(1-t)a+tb: \ t\in\mathbb{R}\right\}=\emptyset.$$ ...
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41 views

Possible explanation for a terminology confusion

In the usual sense, given two spaces $X,Y$ and a family of functions $F$ which map $X$ into $Y$, we say $F$ separates points if for any distinct $x_{1,2}\in X$ there exists at least one $f\in F$ such ...
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179 views

Extension of continuous linear functionals from a closed subspace to the whole locally-convex space

Consider a Hausdorff, real (but I believe this not to be relevant), locally-convex topological linear space $X$ and a closed linear subspace $Y$. Let $f \in Y^*$ (the topological dual). I want to show ...
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377 views

Proof of the Banach-Alaoglu theorem

I have some doubts about the point of this proof of the Banach-Alaouglu theorem: "The cloesed ball $B_{E'}:=\lbrace u \in E' : \left \| u \right \|_{E'} \leq 1 \rbrace$ of dual $E'$ of a normed space ...
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61 views

discrete convexity arising in a simple discrete optimization problem

Let $S$ be a fixed integer satisfying $S \ge 1$, let $a$ range over the integers between $1$ and $S$ inclusive, and for $i = 1, \dotsc, a$, let each $x_i$ range over the nonnegative integers, such ...
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84 views

Metrisability of locally convex spaces/weak topology

The weak topology of a Banach space $X$ is the locally convex topology associated to the family of semi-norms $$ p_f(x)= |f(x)|, \qquad f\in X^*. $$ If $X^*$ is separable, it obviously suffices to ...
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465 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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53 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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24 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 $\textrm{...
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32 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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47 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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21 views

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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231 views

Proof of convexity of a quadratic function

I have the next problem: If $f(x)$ is a quadratic function with n variables: $f(x) = 0.5$$\mathbf{x}^T$$A$$\mathbf{x}$$+$$\mathbf{b}$$^T$$\mathbf{x}$$+$$\mathbf{c}$ were $A$ is a symmetric matrix ...
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72 views

Preserved properties through continuous linear maps

I just looked at the fact (at least according to Definition 2.8.1. in Distribution Theory by Friedlander et al.) that for $K_0\subseteq{\bf R}^{n(0)}$ compact, $\Omega_1\subseteq{\bf R}^{n(1)}$ open ...
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25 views

Regarding embeddings of locally convex spaces

If $f:E\rightarrow E'$ is a linear embedding of locally convex topological vector spaces, and $A\subseteq E$ open and convex, can we always find $A'\subseteq E'$ open and convex sucht that $f(A)=f(E)\...
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52 views

How to show that the vertices of a convex hull are given by these specific subsets…

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

($\forall x\in E_1 \exists y\in E_2: f_1(x_1)=f_2(x_2))\ \Longrightarrow \ x \mapsto y$ continuous

I'm reading through some notes on functional analysis where on a couple of occasion we where in the following setting: $E_1$ and $E_2$ are two Frechet spaces and $f_i:E_i \rightarrow F$ two ...
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17 views

Detecting ray cross after hit on a convex object

I have a hw question im struggling to solve - Any guidance will be appreciated.
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51 views

How to detect reflexivity of the closure

Consider the space of continuous bounded functions on a bounded interval. Its closure for the Lebesgue $L_p$ norm is reflexive when $1 < p < \infty$, but it is not reflexive for $p = 1$. How ...
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367 views

Relative interior inclusion of convex sets

Is relative interior(C-D) = relative interior(C) - relative interior(D) where C and D are nonempty convex sets. If so please give the proof If thats not the case could you give a counterexample ...
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156 views

Convex Sets and Linear Subspace

Consider the set $\{(x,y) \in \Bbb R_+ \times \Bbb R \text{ s.t. } y\leq \ln x - e^x\}$. This set is: A) A linear subspace of $\mathbb R^2$ B) Convex C) Convex & a linear subspace of $\mathbb ...