# Tagged Questions

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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### Reflexive Banach space: Boundedness of subset implies weak compactness. Closed or not?

Claim:In a reflexive Banach space, the weak compactness of a subset is equivalent to the boundedness of the subset. But there is no guarantee that the bounded subset would even have its sequences ...
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### Finding a compact set containing the unit ball in a normed space

I would like to show that there is a compact set $K \supset \{ \Vert x \Vert \leq 1 \}$ in a general normed vector-space $X$, but I have no clue how to do it. Or is it maybe possible to have a finite ...
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### Duality in finite-dimensional normed spaces

Suppose we endow $\mathbb{R}^n$ with a norm $\|\cdot\|$; call such a normed space $X$. Then, as a vector space, the dual space $X^*$ is also $\mathbb{R}^n$. Let $x\in X$ and $f\in X^*$. Consider the ...
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### Will the problem right?

Let $X$ be a normed space and $Y_1$ , $Y_2$ complete spaces and exist two injective linear application $f_1:X \to Y_1$, $f_2:X\to Y_2$ such that $\text{closing } f_1(X)=Y_1$, \$\text{closing } ...