# Tagged Questions

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### Prove that $(A(K), ||$ $||_{\infty})$ is a Banach space. [duplicate]

Define $A(K) = \{f : K \rightarrow \mathbb{R}$ $| f$ is continuous$\}$. $K$ is compact. Prove that $(A(K), ||$ $||_{\infty})$ is a Banach space. Since a Banach space is complete then every Cauchy ...
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### Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
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### Compactness criterion

I have this compactness criterion and I want to apply it, but I don't know what I must write to see if (a) is satisfied and also for (c)? For a subset $H\subset\mathcal{BC}(\mathbb{R},Y)$ to be ...
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### Compactness in $\mathbb{R}^{X}$

I'm reading a book chapter on weak topology, where the author identified the collection of all real functions on an abstract space $X$ with $\mathbb{R}^{X}$. I find it difficult to make sense out of ...
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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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### Completeness/Compactness of a subset in a normed linear space

Let $(X,\|\cdot\|)$ be the normed linear space consisting of the sequences $a=(a_n)_{n=1}^{\infty}$, for which the corresponding series $\sum_{n=1}^{\infty} a_n$ converges absolutely, with norm ...
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### Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
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### Arzela-Ascoli net question

Let $X$ be a compact metric space. Let $C(X)$ denote the space of real-valued continuous functions on $X$. A commonly given corollary to the Arzela-Ascoli theorem is: Proposition: If $f_n$ is an ...
### $X$ normed linear space separable $\Longleftrightarrow$ $\exists K \subset X$ compact s.t. $\overline{ \text{span}\{K\}}= X$
Let $X$ be a normed linear space. Show that $X$ is separable if and only if there is a compact subset $K$ of $X$ for which $\overline{ \text{span}\{K\}}= X$ I can't figure out how to solve this ...