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).

321 questions
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Question related to Fréchet space

I would like to know if the following property that is true for Banach spaces, it holds for Fréchet space as well. The question is: Let $X$ be a Banach reflexive space and $Y$ be a Fréchet space. ...
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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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Is $A = \{ (z_1 , z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2|\}$ an absorbing, balanced and convex set?

Is $A = \{ (z_1 , z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2|\}$ an absorbing, balanced and convex set? I am new in StackExchange I from Colombia, because I don't write English very well. Edited from ...
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Infinite-dimensional spaces for which strong and weak topologies coincide

For a dual pair $\langle X, Y \rangle$ of vector spaces $X$ and $Y$ denote by $\sigma(X,Y)$ the weak topology and by $\beta(X,Y)$ the strong topology on $X$. Consider the following known statement (S)...
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Relations between different definitions of polar topologies

Let $\langle X, Y \rangle$ be a separated dual pair of (real) vector spaces. A non-empty family $\mathcal{A}$ of non-empty subsets of $Y$ is called polar if every $A \in \mathcal{A}$ is bounded (w.r....
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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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Mackey topology vs. uniform convergence on weakly compact convex sets

Let $X$ be a locally convex Hausdorff space and $X'$ its dual space. By the Mackey-Arens theorem, there is a finest locally convex topology $\tau$ on $Y$ such that $(Y, \tau)' = X$. $\tau$ is called ...
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Boundedness in locally convex space

Consider a vector space $X$ equipped with a separating countable family of seminorms $(p_i)_{i\in\mathbb{N}}$. I'm told that the boundedness of a set in $X$ with respect to each one of the seminorms ...
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Continuous Map for the Compact Open Topology

Suppose I have a map $\Phi:\mathbb{R}\rightarrow C_b(\Omega)$ where the right hand side is equipped with the compact open topology $\tau_{co}$. How to show that such a function is continuous with ...
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State space of $C([0,1])$

Consider the $C^*$-algebra $A := C(X)$, where $X$ is a compact Hausdorff space. Denote by $S(A)$ the state space of $A$. In the weak$^*$ topology this is a convex compact set such that $S(A)$ is the ...
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Locally Convex Topology on $C_b(\Omega)$

Let $\Omega$ be a locally compact Hausdorff space and look at $C_b(\Omega)$. Then we can define a topology $\tau$ on this space by being the finest locally convex topology agreeing with the compact ...
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Continuous seminorms

If I have a locally convex vector space $S$ equipped with a countable family $(p_i)_{i \in I}$ of seminorms, is it correct that the topology remains unchanged if I add an extra seminorm $q$ satisfying ...
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Continuous semi-norms on subspace

Suppose $X$ is a locally convex topological vector space, let $P$ be the set of all continuous semi-norms on $X$. Suppose $M$ is a subspace of $X$, denote $P|_M$ as the set of semi-norms in $P$ ...
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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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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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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 arcwise-...
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 ...