# Questions tagged [topological-vector-spaces]

The study of vector space with a topology which makes the maps which sums two vectors and which multiply a vector by a scalar continuous. It's a natural generalization of normed spaces.

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### Properties of the neighborhoods of $0$ in the cofinite topology

Let $E$ be a vector space with a non countable number of elements. Define $$\mathscr{F}=\{U \subset E:0 \in U \hbox{ and } \#E\setminus U<\infty\}.$$ It's easy to see that $\mathscr{F}$ defines a ...
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### On a result of Mazur about convergence in locally convex spaces

My question is about the following result from Simon (2011) (Theorem 5.3). Let $X$ be a locally convex space and $Y$ its space of continuous [linear] functionals. Let $\{x_n\}$ be a sequence in $X$ ...
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### Can any two points in an open and connected subspace of a locally convex t.v.s. be connected by a continuously differentiable path?

Can any two points in an open and connected subspace of a Hausdorff locally convex space be connected by a continuously differentiable path? Some known related facts: An open and connected subspace ...
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### Existence of a continuously differentiable function with prescribed properties

Given a sequence $t_n \rightarrow 0$ of pairwise distinct positive real numbers and a sequence $x_n \rightarrow x$ in a Hausdorff locally convex space $X$, is there a continuously differentiable ...
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### Conclude that the following topological vector space is not first-countable

Suppose that we have a normed vector space $(X,\|\cdot\|)$. We endow $X$ with a (locally convex) topology $\tau$ such that any $\tau$-convergent is norm bounded. Suppose that there exists a countably ...
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### How do I show that for an open, nonempty subset $A$ of a topological vector space $V$ is convex?

I'm trying to prove the following: $V$ topological vector space. $A,B\subseteq V$ nonempty, open subsets. Suppose that a hyperplane $H\subseteq V$ separates $A$ and $B$. Show $H$ strictly separates ...
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### Closedness of the sum of two cones

Consider two closed convex cones $K_1$, $K_2$ in a topological vector space. It is known that, in general, the Minkowski sum $K_1 + K_2$ (which is the convex hull of the union of the cones) need not ...
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### Finding a topology for a vector space $X$, such that $X$ is a topologic vector space and the topologic and algebraic dual spaces are the same
Given is a vector space $X \neq \{0\}$. I should find a topology such that $X$ is a topologic vector space and the topologic $X'$ and algebraic $X^*$ dual spaces are the same. My idea is to use the ...