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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255 views

Dual of the linear map

I know the meaning of dual of linear map in inner product spaces, also it is defined in Banach space[ Rudin functional analysis]. What is the definition of Dual of linear map if vector space are ...
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117 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. ...
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464 views

Inductive Limit of directed locally convex Frechet Spaces

Let $\Phi=\bigcup_{i\in \mathbb{N}}\Phi_i$ be the inductive limit of an upwardly directed set of countably-seminormed spaces (i.e. the locally convex topology is given by a countable family of ...
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297 views

Locally convex topological vector space

In $C(\mathbb{R})$ space we define two families of seminorms: $p_x(f)=|f(x)|$ where $ x\in\mathbb{Q}$ and $q_x(f)=|f(e^x)|$ where $ x \in \mathbb{R}$ I have to check if above families induce locally ...
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173 views

On convex hulls and intersections of chains of compact sets

Let $V$ be a topological vector space, let $\{ C_i \}_{i \in I}$ be a set of compact subsets of $V$ which forms a chain with respect to inclusion. For now, assume the following stronger properties: $...
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281 views

Can I decompose a compact set in a finite number of convex set?

My problem is in a finite-dimensional space. I look at $\mathcal{X}$ the support of a function $f$, that is continuous and has bounded support. \begin{eqnarray} \mathcal{X}_o & = & \{x \in \...
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403 views

affine set definition equivalence

How to show the following definitions are identical for an affine space: $C = p + W$ where $W$ is a subspace $p$ is a vector in $\mathbb{R}^n$, and $\lambda a + (1-\lambda) b$ is in $C$ for any $a$ ...
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directional derivative of convex function sublinear proving that fact

How can we show that the directional derivative of a proper convex function on $\mathbb{R}^n$ is sublinear? Thank you!
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171 views

affine function definition

If we define the affine function as $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for every $x,y \in R^d$ and $\lambda \in R$ How to show that it is equivalent to the definition ...
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1answer
52 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) $X$ ...
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98 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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27 views

Is this map continuous?

Let $(E,\langle\cdot\;,\;\cdot\rangle)$ be a complex Hilbert space. For $M\in\mathcal{L}(E)^+$ (i.e. $M^*=M$ and $\langle M x\; |\;x\rangle\geq0$, for all $x\in E$). We define the semi-inner product $...
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153 views

First countability, nets and sequences in the weak topology.

According to this book (see print below) the weak topology (as well as the weak* topology) is first countable. However the weak topology should be first countable only in the finite dimensional case. ...
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179 views

Why sequential continuity from $E$ to $E'$ implies continuity?

Let $(E,\|\cdot\|)$ be a separable Banach space. Let $E'$ be the topological dual of $E$ equipped with the weak* topology $w^*$. I read that a certain linear operator $J:(E,\|\cdot\|)\to (E',w^*)$ is ...
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54 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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196 views

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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342 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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134 views

Continuity of the dual product

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 on $X\times X^*$, mainly because ...
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176 views

Counterexample for bounded subsets in a Frechet space

I do not know if anyone has asked about this before: Does anyone know an example of an unbounded subset of a Frechet space which have finite diameter? I saw this in Conway (A course in functional ...
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1answer
116 views

Quasi-Convexity

Can I get the conclusion that the function of matrix $P$ and $Q$ \begin{equation} \mathrm{tr}\left( PQ\right) \end{equation} is a quasi-concave function for $P>0$, and $Q>0$? It is true for ...
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1answer
15 views

Characterisation of the locally convex spaces which are sequentially dense in their completion

Is there a characterization of the locally convex spaces with the property that they are sequentially dense in their completion. In other words, under which conditions on a locally convex space $E$ is ...
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1answer
56 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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1answer
84 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 ...
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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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54 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
76 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_{\...
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1answer
27 views

Separating the closed convex envelope of a compact set in Frechet space by a countable family of continuous linear functionals

A space is called Frechet if it is complete metrizable locally convex space. Suppose $Y$ is a Frechet space, and $K$ is a compact subset of $Y$. We let $V$ denote the closed convex envelope of $K$. Is ...
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1answer
130 views

Convex sets in Linear topological spaces

A (real) linear topological space is a real linear space (vector space) $Ε$ with a Hausdorff topology such that: I) vector addition is continuous II) scalar multiplication is continuous For $x$ ...
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1answer
90 views

The notation of weak-star topology on the second dual of a locally convex space

I don't understand some notation. I know that if $\mathscr{X}$ is a LCS, then $(\mathscr{X}^*,\text{wk}^*)^* = \mathscr{X}$, where $\text{wk}^*$ denotes the weak-star topology on $\mathscr{X}^*$. ...
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1answer
89 views

Is a sequentially continuous map $f:E'\to E'$ continuous?

I have read that for a separable complete locally convex space $E$, any sequentially continuous linear map $f:E'\to \mathbb{K}$ is continuous (where $E'$ is equipped with the weak*-topology). Is ...
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87 views

Approximation property for $C^k([0,1]^m)$

This must be well-known, of course, so excuse me my ignorance. I think, the Banach space $C^k([0,1]^m)$ (of $k$ times smooth functions on a hypercube $[0,1]^m$) must have the approximation property. ...
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1answer
37 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
50 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 $\textrm{ex}(K)$...
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94 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 $y_n\in X\setminus V$...
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37 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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1answer
411 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 ...
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1answer
164 views

Topological modules and relative homological algebra.

This question might be a bit dumb but I'm tired right now and this is just going over my head at the moment, in "The homology of Banach and topological algebras" Helemskii said that relative ...
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250 views

Is the algebraic interior relatively open in a closed convex set?

I am struggling with proving or disproving the following: Let $X$ be a locally convex space and let $C\subset X$ be closed convex with non-empty algebraic interior which we denote by $C^i$ (recall ...
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1answer
67 views

Condition under which a locally convex topological vector space becomes a normed linear space

Is this true that a locally convex topological (Hausdorff) vector space becomes a normed space when its local base has only one element, so only one Minkowski functional and so only one seminorm and ...
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253 views

Algorithm for determining points from given dataset that are within a convex hull defined by a subset of original dataset

Having a dataset of x,y,z with n points like: ...
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1answer
84 views

convex hulls of convex sets plus a point

In a real vector space, consider S and T to be disjoint, nonempty, convex subsets of the vector space and let a point x lie outside either set. How would I prove the following: co({x} union S) is ...
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12 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
52 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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15 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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30 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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22 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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41 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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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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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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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 $x\in M$? ...