# Tagged Questions

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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### If $a$ and $b$ commute in a $C^*$-algebra and $a$ is normal, then $f(a)$ and $b$ commute for any continuous $f$

I'm trying to find a way to demonstrate the following: Let $(A,*,\|\cdot\|)$ be a unital $C^*$-algebra. If $a,b\in A$ commute and $a\in A$ is normal (i.e. $a^*a=aa^*$), then for every continuous ...
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### Open maping theorem. Completeness assumption are important [duplicate]

The open maping theorem between banach spaces says. Let $T:X\to Y$ be a linear,continuous and surjective map between the banach spaces $X,Y$ then $T$ is an open map. I need examples to show that the ...
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### Banach spaces over complete fields with their own absolute value

Let $F\hspace{.03 in}$ be a field, and let $E\hspace{.03 in}$ be an ordered subfield of $F$. Let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| \: : \: F \: \to \: E \;\;$ be such that for all ...
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### Continuous mapping on unit ball such that $T(\Bbb x_0)=0$

got a question from a course in functional analysis. " Let $T:\{\Bbb x\in\Bbb R: ||\Bbb x||\leq 1\}\to\Bbb R^n$ be a continuous mapping. Moreover assume that $\langle T(\Bbb x),\Bbb x\rangle>0$ ...
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### Showing infinite direct sum of Banach spaces with a certain norm is a Banach space

Given a family $(A_{\lambda})_{\lambda\in\Lambda}$ of Banach spaces, let $\bigoplus_{\lambda}A_{\lambda}$ be the set of all $(a_{\lambda})\in\prod_{\lambda}A_{\lambda}$ such that ...
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### Weak* sequentiality

Suppose we are given a Banach space $E$ such that weak* compact subsets of $E^*$ are weak* sequentially compact (for example this happens when $E$ is separable). Does it follow that if $A$ is a subset ...
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### $C_0^\infty(0,T)\cdot V$ dense in the Bochner space $L^2(0,T;V)$

Let $V$ be a Banach space and $(0,T)$ a time interval. Consider the space $C_0^\infty(0,T)$ of infinitely often differentiable functions with values in $\mathbb R$ and compact support in $(0,T)$ and ...
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### Is space of Dirac measures Banach?

Is the space of all Dirac measures on a set $\Omega$ Banach? With the total variation norm. I don't know what convergence means in this norm.. I mean how do I even think about it.
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### Is there a non-reflexive Banach space which is strictly convex?

I just come up with the fact that a space being strictly convex, does not implies it is reflexive (at least I never saw a proof of it). How can one construct a example of a non-reflexive Banach ...
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### Convergence of functionals and weak convergence

I consider a Banach space $V$ with its dual $V'$. I had a sequence of functionals $\{f_k\}_{k\in \mathbb N} \subset V'$, and I wanted to show (strong or norm) convergence of $f_k \to f \in V'$. I ...
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### Image of the tensor product of strict maps of Banach spaces

Let $f:A\to C$ and $g:B\to D$ be bounded linear maps of Banach spaces with closed image. Will $f\widehat{\otimes}g:A\widehat{\otimes}_\pi B\to C\widehat{\otimes}_\pi D$ also have closed image? What ...
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### Equivalent conditions for weak and weak-$*$ convergence

Say $E$ is a normed space over the field $\mathbb K$ ($\mathbb R$ or $\mathbb C$) and $E^{*}$ its dual space. The notations for weak and weak - $*$ convergence are $x_{n} \xrightarrow{w} x$ and ...
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### The Banach space $c_0$ is $C^{\infty}$-smooth.

In this paper, J. Eells defines this notion of $C^r$-smoothness for Banach spaces: A Banach space $E$ is $C^r$-smooth, $r \geq 0$, if there exists a nontrivial (that is, nonzero) $C^r$ function ...
### Proving the $l_p$ space is complete.
I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...