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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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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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 $U'$...
- 3,615
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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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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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Exercise about convex hull and weak convegence
Suposse $X$ is a normed space, $(x_k)_{k \in \mathbb{N}}$ a sequence in $X$, $x \in X$ such that $x_k \underset{weakly}{\rightarrow} x$.
Let $co$ denote the convex hull.
Show that, there exists $y_k \...
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Complete locally convex topological vector spaces are not stable under extension
I've heard that complete locally convex topological vector spaces are not stable under extension. However, I don't know of any example. What would be an example of a complete topological vector space $...
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Question about weak and pointwise convergence
I have a question about weak topology.
Definition: If $X$ is a LCS, the weak topology on $X$, denoted by "wk" or $\sigma(X,X^*)$, is the topology defined by the family of seminorms
$\{p_f : ...
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Understanding the definiton of weakly open sest and weak convergence
I am learning about weak and weak* topology. In the book I am reading the following is mentioned
Definiton (weak topology) If X is a LCS, the weak topology on X, is
the topoloty defined by the family ...
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Difference between the weak and weak* topology (using seminorms to define the topologies)
A few days ago, I was interested in the weak topology and the fact that the weak topology is the coarsest topology such that $f:X \rightarrow \mathbb{K}$ is continuous.
(How to show that, the weak ...
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Lemma about Minkowski Functional in topological vector spaces
I am trying to prove the following lemma:
Suppose $X$ is a topological vector space.
Show that if $S \subseteq X$ is a convex (open) neighborhood of $0$ there exists a non-negative continuous ...
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How to show that, the weak topology is the coarsest topology such that all $f:E \rightarrow \mathbb{K}$ are continuous?
Let E be a normed space, $E':=\{f:E \rightarrow \mathbb{K}| f \text{ is continuous and linear}\}$. Define $p_f(x):=|f(x)| where f \in E'$ and $x \in E$. Consider the family of seminorms $\mathcal{P}=\{...
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