# Tagged Questions

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### Closure of convex set [on hold]

Is the closure of a convex set (in a normed vector space) itself convex? I can't think of a counterexample!
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### Linear bijection non-preserving Hausdorff propery

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more ...
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### Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
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### Show that no topological vector space is bounded.

I am studying the concept of topological vector spaces in Grubb's Distributions and Operators. A vector space $X$ (over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$) is called a topological vector ...
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### Prove that a given subspace of $C[-1,1]$ with $L^2$ norm is closed

Let $H= C[-1,1]$ with $L^2$ norm and consider $G=\{f \in H \mid f(1) = 0\}$. Show that $G$ is a closed subspace of $H$. I've been trying to prove this for a while but i can't establish that given ...
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### Looking for proof that an open set in vector space contains the sum of two open sets.

Problem: To show that, in a topological vector space, for a given neighborhood of zero $W$, there exist two neighborhoods of zero, $V_1$, $V_2$, whose sum is contained in the first neighborhood, ...
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### Compact set - prove that supremum is actually maximum

There is a compact set $K$ in $\mathbb{R}^n$. The diameter of this set is defined as follows: $D = \sup\limits_{x,\, y\, \in K}\|x-y\|$. I need to prove there are two vectors $a,b$ in $K$ such ...
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### Smallest/Minimal bases of a topological space

The smallest possible cardinality of a base is called the weight of the topological space. I was wondering if all minimal bases have the same cardinality, and if every base contains a subset whose ...