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# 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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### 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 ...
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### 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 ...
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### 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 $x\in M$? ...
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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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### A convex function with a non-empty domain interior in a non-barreled locally convex space

Given a (Hausdorff separated) locally convex space $X$ what can we say about a proper convex function $f:X\to\mathbb{R}$ whose domain $\emptyset\neq D(f):=\{x\in X\mid f(x)<+\infty\}$ has a non-...
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### About continuity on the space of measures.

Consider a separable, metrizable and locally compact space $\mathrm{X},$ that is not compact. Define $\mathrm{M}_{\mathbf{C}}(\mathrm{X})$ to be the set of complex measures on $\mathrm{X}$ and regard ...
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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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### What are weakest conditions on a vector space that allow defining dual (adjoint) operator?

I have an infinite dimensional topological vector space $V$ and its dual $V^*$ endowed with a sesquilinear form $$\phi:V^*\times V\to \mathbb{C}: (u,v)\to \phi(u,v) , \qquad u\in V^* , v\in V$$ I've ...
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### Closed graph theorem for operator topologies - Do operator topologies yield Fréchet spaces?

Consider the SOT and WOT operator topologies on $B(H)$, the bounded operators on a hilbert space $H$. I'm interested in the properties of the topological vector spaces induces on $B(H)$. Are they ...
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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 ...
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### Prove limit exists for an almost monotonic bounded function, involves convexity and square integrability

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 $\vert x(t)\vert \leq x_{\max}<\infty$ ...
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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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### 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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### 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 ...