# 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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### Proof that convergence in probability is not locally convex

On Wikipedia, it is stated that the topology induced by convergence in probability is not locally convex. However, no proof nor reference is provided, and I couldn't find anything on Google. I think ...
36 views

### Convex hull of bounded set is bounded. Is my proof right?

I want to prove the following theorem. Let $X$ be a topological vector space. If $X$ is locally convex then the convex hull of every bounded set is bounded. My proof: If $A$ is bounded, for every ...
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### Vector space topologies stronger than the strongest locally convex topology

Every real vector space has the strongest locally convex topology, which is the topology generated by all the convex sets whose intersection with every line is an open interval. What about topologies ...
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1 vote
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### An interesting family of seminorms $\mathcal F$ and comparison between the topology generated by this seminorms and the Weak Operator Topology.

I am learning functional analysis and I am stuck with the following questions from Strong Operator Topology and Weak Operator topology on $\mathcal B(H)=\{T:H\to H:T$ is Op-Norm continuous,linear $\}$....
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### Do "halves" of open sets exist in locally convex vector spaces?

Let $V$ be a locally convex Hausdorff topological vector space (over $\mathbb{R}$) and let $U\subseteq V$ be an open neighbourhood of the origin. Does there always exist another open neighbourhood $U'$...
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### convex combination of probability measures [closed]

$\left( \Omega,\mathcal{A} \right)$ is a measurable space and $\mu,\nu$ are probability measures on it. Prove any convex combination of $\mu$ and $\nu$ is also a probability measure on this space. ...
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### Why is closedness crucial for a barrel set to be a neighborhood of the origin in a Banach space

Apology for asking highly related but subtly different questions within a very short amount of time. Let $X$ be a real Banach space, and $A\subset X$ be balanced, convex, and absorbing. Two facts are ...
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### Minkowski functional as a norm

Related to my previous question. Let $X$ be a real Banach space, and $A\subset X$ a balanced, convex, absorbing set that is bounded. Then $(X, p_A)$, where $p_A$ is the minkowski functional, is a ...
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1 vote
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### Schauder bases in inductive limits

Assume $X_n$ is a family of nested Banach spaces (i.e. $X_n\subset X_m$ whenever $m>n$ and the inclusion map is continuous) and denote by $X$ the inductive limit $\lim X_n$. Assume moreover that $X$...
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### Is every extreme point in a compact convex set contained in a defining supporting hyperplane?

Let $K \subseteq X$ be a compact convex subset of a locally convex space $X$. Let $k \in K$ be an extreme point. Question 1: Does there exist a supporting hyperplane of $X$ containing $k$? I think the ...
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### Convexity of a connected, compact, and locally convex set in $\mathbb{R}^n$

Is a compact, connected, and "locally convex" set in $\mathbb{R}^n$ convex? Here I mean a space $A$ locally convex as: For any point $x\in A$, there exists a neighborhood $U$ of $x$ s.t. $U$...
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### Understanding equivalent definitions of locally convex spaces

As stated in the title, I am trying to make sense of the equivalent definitions of locally convex spaces. Especially, I am confused about the part, where I try to prove, that the topologies coincide. ...
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1 vote
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### Is the weak operator topology equal to $\sigma(L(X,Y), X\times Y^*)$?

I'm asking myself if the weak operator topology is equal to the weak topology $\sigma(L(X,Y), X \times Y^{*},b)$ with  \begin{align} b:L(X,Y)\times (X \times Y^{*})\to& [0, \infty)\\ (T,(x,y') \...
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### $\varphi\mapsto x^\alpha \varphi$ is a continuous endomorphism in Schwartz space

I'm following some notes on functional analysis where it's shown the result presented on the title. The proofs in the notes is not clear to me. Before showing the proof I give some context. The ...
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### $x_\beta \rightarrow x$ in a locally convex space if and only if $\rho_\alpha(x_\beta, x) \rightarrow 0$ for every seminorm $\rho_\alpha$

Let $X$ be a locally convex space with the family of seminorms $\{p_\alpha\}$. I am trying to get a feel for convergence of nets (or sequences) in these spaces aside from the general topological ...
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### Unit semiball of the supremum of seminorms is closed

I don't know how to prove the second part of this exercise (ex. 7.2 in F. Treves, "Topological vector spaces, distributions and kernels"). Let $\mathcal{P}$ be a family of continuous ...
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1 vote
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### Confusion about locally convex (topological vector) spaces

The author in the book I am reading defines a locally convex space as follows. Definition: A topological vector space $(X,\tau)$ is called a locally convex space if there is an index set $A$ and a ...
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### Subbase for initial topology of quotient mappings (locally convex spaces)

Consider a family $\{p_i\}_{i\in I}$ of seminorms defined on $X$ and consider for each $i\in I$ the quotient mapping $q_i:X\rightarrow X/\ker(p_i)$. I know, that each quotient $X/\ker(p_i)$ is a ...
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Let $X$ be an infinite dimensional locally convex space that separates points and $X^*$ its dual. I would like to prove that no $\sigma(X, X^*)$-continuous semi-norm is actually a norm. What I have ...