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).

322 questions
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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 ...
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 $x\in M$? ...