# Tagged Questions

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### closed unit ball in a Banach space is closed in the weak topology

Let $V$ be a Banach space. Show that the closed unit ball in $V$ is also closed in the weak topology. I know this is a consequence of the statement any closed convex subset in $V$ is closed in the ...
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### Question on Baire Property

In reading Banach's book, Theory of Linear Operations, I have a question on the definition of Baire property, or Baire condition in Banach's book. Here is the definition in Banach's book: Definition. ...
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### A vector space with topology generated by a family of typologies each makes it a topological vector space is a topological vector space

Let $V$ be a vector space, and let $(\mathcal F_ \alpha ) _{ \alpha \in A}$ be a family of topologies on V, each of which turning $V$ into a topological vector space. Let $\mathcal F$ be the vector ...
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### Topologies of test functions and distributions

I'm wondering about some of the topological properties of $\mathcal D(\Omega)$ and $\mathcal D'(\Omega)$: I know $\mathcal D(\Omega)$ is not metrizable, so not first countable (right?). However, my ...
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### a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
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### Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
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### $\ell^{\infty}(\mathbb N)$ is not a separable space

I have to prove that $\ell^{\infty}(\mathbb N)$ is not separable. My attempt Consider a SUBSET $V$ of $\ell^{\infty}(\mathbb N)$ consisting of bounded sequences that have only $0$, $1$ entries, e.g. ...
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### The dimension of the real continuous functions as a vector space over $\mathbb{R}$ is not countable?

This question is out of curiosity. I first attempted a web crawl for this answer but was befuddled when Google didn't turn up the result after a couple of tries. If anyone has a reference, I'd be ...
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### When does a dense subspace destine the weak topology?

Let $E$ be a locally convex space, let $E^{\prime}$ be its continuous dual space and let $F$ be a subspace of $E^{\prime}$ which is dense with respect to the strong topology on $E^{\prime}$ (i.e. the ...
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### Topology induced by Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
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### Existence of non-locally constant function

Let $K$ be a compact Hausdorff space without isolated points and let $x \in K$. Does there always exist a real-valued continuous function that is not locally constant near $x$? If $K$ is metrizable ...
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### Ultraweak topology on Banach spaces

If $X$ and $Y$ are Banach spaces with $Y$ reflexive, then the space $\mathcal{B}(X,Y)$ of bounded operators from $X$ to $Y$ is the dual of the projective tensor product of $X$ and $Y^{*}$. As in the ...
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### Topology of space of continuous functions

Let $X,Y$ be topological spaces and let $C^0(X,Y)$ be the set of continuous functions between them, endowed with the compact-open topology. I am interested in the following kind of questions: What ...
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### Can natural quotient map between Banach spaces be closed?

Let $X$ be a Banach space and $M$ be closed subspace of X, and let $q:X\to X/M$ be the natural quotient map. I know that $q$ is an open map. I wish to find an example of $X$ and $M$ such that $q$ is ...
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### If a neibourhood of the origin shrinks to the origin then its closure also shurinks to the origin?

In the proof of the open mapping theorem (theorem 2.11) in his Functional Analysis, Rudin states ... $y_{m+1} \rightarrow 0$ as $m \rightarrow \infty$ (by the continuity of $\Lambda$)... In the ...
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### Locally-compact function spaces?

I ask this question out of curiosity, not a specific need. Euclidean spaces and manifolds. Are there examples of locally compact function spaces? Could (some?) Sobolev spaces be locally compact?
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### Continuity of covariance kernels

Let $I$ be a locally compact Hausdorff (LCH) topological space. Let $c : I \times I \to \mathbb R$ be a covariance kernel, that is, a symmetric, nonnegative-definite function. Does it follow that $c$ ...
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### Weak Compact and separable sets

Is true the following statement? Se $(X\|\cdot\|)$ a Banach espace, and $K\subset X$ a convex, weakly compact and separable set. Let $x_{n}$ a sequence in $K$. Thus, given any $\epsilon >0$ there ...
I'd like to prove that: Given a Hilbert space H and S a closed subespace, $S \subseteq H$, the projection $P_{S}:H \to S$ is continuous in the weak topology. I have tried the following. ...
I have a norm that works in function space and that is $‖∙‖_{sup}:C([0,1])→R$, $‖∙‖_{sup}:=sup${$|f(t)|$}. I need to show that the metric is complete. So I need to show that every Cauchy sequence of ...