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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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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Why is $e \in C$ (a commutative subalgebra of $A$)?

I am reading the following proposition: Here $e$ represents the identity element of $A$ and $\sigma_C(x)$ and $\sigma_A(x)$ denote the spectrum of an element $x$ in $C$ and $A$ respectively. ...
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How do I prove the Local intersection property in the example(Economics)

$T$ is said to have local intersection property if for each $x\in X$ with $T(x)\neq\emptyset$, there exists an open neighborhood $N_{x}$ of $x$ such that $\cap_{z\in N_{x}}T(z)\neq\emptyset$. ...
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It is always possible to define a topology in a vector space endow with a semi-norm?

If $(X,\|\cdot\|)$ is a semi-normed vector space. It is always possible to define a topology on $X$? If it is true What is the definition of a closed subspace of $X$ with respect to $\|\cdot\|$? ...
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Elementary proof that every compact convex subset of a normed vector space has an extreme point

I've shown that the following result is valid for $V=\Bbb R^v$: If $K\subset V$ is compact and convex and $V$ is a $v$-dimensional Banach space, then $K$ has at least one extreme points. The proof ...
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Is there a simple proof that every compact convex $K\subset \Bbb R^n$ has at least one extreme point?

Can someone give me a simple proof that every compact convex $K\subset \Bbb R^n$ has at least one extreme point? I'd found this one but couln't understand it: Convex compact set must have extreme ...
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Showing that $\text{int}(\text{ch}(x_1, …, x_k))\subset\text{int}(\text{ch}(x_1, …, x_n))$

I am studying the book Convexity: An Analytic Viewpoint by B. Simon and I am not sure if understood a passage in the following proof (page 124): Proposition 8.7 $\text{ch}(e_1, ..., e_n)$ always ...
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Help on a passage

I am trying to understand the following proof, which seems to be simple, but I think I am missing something. I couldn't find the open interval about $L(y)$. Lemma 4.2: Let $A$ be an open convex set ...
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Over the separation of convex sets in a Banach space

I am working with Banach spaces and aim to prove the following property: If $A$ and $B$ are disjoint convex sets of a Banach space $X$ with $A$ open, then $A$ and $B$ can be separated, that is, ...
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Convex hull of extremal points

Let $E$ be a locally convex Hausdorff space and $X$ a compact convex set. Is it true that $\overline{conv}(\overline{ex(X)})=\overline{conv}(ex(X))$? Here $conv(X)$ is the convex hull of the subset $X$...
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Barreled topology finer than the weak-star topology

I have an infinite-dimensional (Hausdorff separated, non-metrizable) locally convex space $(X,\tau)$ with topological dual $X^*$. My questions are: Under what conditions is there a barreled ...
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Quotient topology defined by seminorms

Please Prove this: If $p$ is a seminorm on $X$, where $M$ is a linear manifold in $X$ and ${p}_M:X/M \rightarrow[0,\infty)$ is defined by $p_M(x+M)=\inf\{p(x+y): y \in M\}$ then $p_M$ is a seminorm ...
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Is every seminorm induced by a linear operator into a normed space?

I'm reading an analysis textbook chapter on convex topological vector spaces, and there is this statement that (one of) the most common way(s) to define a topology on a vector space $X$ is by ...
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Why is $\bigcup\limits_{|c| \leq 1} cU$ convex?
I have a question about proposition in this set of notes (https://www.math.ubc.ca/~cass/research/pdf/TVS.pdf) on topological vector spaces. We are in a complex vector space $V$. A subset $S$ is said ...
Let us consider the dynamics $\dot{x}(t)=-\gamma(t)y(t)\left(y(t)+m(t)\right)$ where $\gamma(t)\in [0,1]$ is a convex function that gaurantees the bound $\vert x(t)\vert \leq x_{\max}<\infty$ ...