# Tagged Questions

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### Properties of compact set: non-empty intersection of any system of closed subsets with finite intersection property

Let $X$ be a Hausdorff topological vector space. Let $C$ be a nonempty compact subset of $X$ and $\{C_\alpha\}_{\alpha \in I}$ be a collection of closed subsets such that $C_\alpha \subset C$ for each ...
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### Compactness in Infinite Dimensional Vector Spaces

Show that, in an infinite dimensional normed space $(V,\|\cdot\|)$, the closed ball of radius $2$ $$B_2:=\{x\in V:\ \|x\|\leq2\}$$ is not compact. I suspect I am not understanding what is going ...
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### Is $C([0,1])$ a compact space?

Is $C([0,1])$ (I guesss with the max-norm) a compact space? I have to know that because I want to apply Arzela Ascoli.
### Bounded and compact sets in a subspace of $\mathbb R^{\mathbb N}$
Let $$X= \{u=(u_1, u_2, \ldots): u_n \ne 0 \text{ only for a finite number of terms}\}\subseteq\mathbb R^\mathbb N,$$ with the topology inherited from $\mathbb R^\mathbb N$ (the "pointwise ...
### If $A$ and $B$ are compact, then so is $A+B$.
This is an exercise in Chapter 1 from Rudin's Functional Analysis. Prove the following: Let $X$ be a topological vector space. If $A$ and $B$ are compact subsets of $X$, so is $A+B$. My guess: ...