Tagged Questions

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Every Bounded set contained in a Compact set

In a general metric space, is every bounded set contained in a compact set?
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Does a barrel contain a neighborhood of $0$? [closed]

Suppose $X$ is topological vector space which is of the second category in itself. Let $K$ be a closed, convex, absorbing subset (a barrel) of $X$. Prove that $K$ contains a neighborhood of $0$.
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Example of a sequence of functions

Construct an example of a sequence of functions $(f_n)$ defined on $[0,1]$ such that $f_n$ converges pointwise to $0$ and for every sequence of numbers $(a_n)$ that tends to $\infty$, sequence ...
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Question on the proof that $C(\Omega)$ is a Frechet space

I am using Rudin's book on Functional Analysis. I am studying the proof that the space $C(\Omega)$ of continuous functions on an open set $\Omega \subseteq \mathbb{C}$ is a Frechet space. I ...
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Does $H(\operatorname{div})$ have a Schauder basis?

Let $\Omega$ an open bounded subset of $\mathbb{R}^n$, $n\in\{2,3\}$, and let $$H(\operatorname{div};\Omega):=\{v\in L^2(\Omega):\operatorname{div}v \in L^2(\Omega)\}.$$ My question is: does ...
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closed subspace of normed vector space

Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.
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We say that $f:[0,1]\to \mathbb{R}$ satisfies Lusin's condition (N) provided $$m(f(B))=0 \quad\mbox{whenever}\quad B\subseteq [0,1] \mbox{ with }m(B)=0$$ where $m$ stands for the Lebesgue measure on ...
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Hahn-Banach Separation Theorem and Bishop's Theorem

I am looking at the proof of Bishop's Theorem on pages 122 and 123 of Rudin's Functional Analysis. The following quote is from the the last two sentences of the proof on pg. 123. "Every continuous ...
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Density problem

$U$ is any open set of $\mathbb{R}$. We known that $C_0^\infty(U)$ is dense in $C^k(U)$. But what about, say $C_0^\infty((0,1))$ in $C^k([0,1])$?
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When is a subset of {0,1} valued borel functions on a standard borel space (polish space) complete (see *) under the pointwise convergence topology?

*In other words, what restrictions on a family F of {0,1} valued borel functions will tell us that the pointwise limit of any net in F is borel. I feel like there must be lots known about this but I ...
Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
Let $M$ be the positive borel measures on a hausdorff topological space $X$, which are finite on compacts sets $--$ i.e. the real cone of radon measures. I am given a definition of a derivative of ...