# 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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### Weak* subseries convergent implies norm convergent?

Suppose $X$ is a Banach space and $X^{*}$ is separable. Suppose that $\sum x_{n}^{*}$ is a series in $X^{*}$ which has the property that every subseries $\sum x_{n_{k}}^{*}$ converges weak*. Show that ...
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### Understanding the relation of weak and weak star toplogy

I'm working with Eberlein- Smulian Theorem fromm the book "Topics in Banach Space Theory". During the proof I have seen that there is used a lot the concept of weak topology and weak star topology. ...
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### Banach space with cardinality bigger than $\mathfrak{c}$.

By using the infromation contained in this post, we have that the cardinality of every Banach space is equal to its dimension, which in turn, is bigger or equal to $\mathfrak{c}$. In my area of study,...
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### Is this space a banach space?

Hi I want to find out whether $l^1$ with the norm $||x||:=sup_n |\sum_{i=1}^{n} x_i|$ is a Banach space. In case that you think that it is a Banach space, just say: It's a Banach space(and then I will ...
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### Supremum over dense subset of banach space

Let $\{x_n\}$ be a countable dense subset of a Banach space $X$. How can I show that $$\sup_{x \in X}f(x) = \sup_{n \in \mathbb{N}}f(x_n)$$ where $f$ is continuous and real-valued??
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### Frechet derivative of compact operator is compact

... this seems to be a well known fact as mentioned in this and in this manuscript. However, I was not able to find a proof or to prove it by myself. So my question is: How to prove this? Any hint ...
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### Proving a Space of Real Valued sequences is Banach.

Theorem: A normed vector space $(V,||\circ||)$ is a banach space if and only if for every sequence $x_n$ in $V$ with the property that $\sum ||x_n||<\infty$ we have $\sum x_n < \infty$. ...
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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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### Banach spaces and Normed linear spaces

Here's a theorem: A normed linear space X is a Banach space iff every absolutely convergent series in X is convergent. How is this possible? I need the proof.
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### How to show that the dual of $(\mathbb{R}^n,\|{\cdot}\|_p)$ is $(\mathbb{R}^n,\|{\cdot}\|_q)$?

I am trying to brush up on my functional analysis and I learn some $L_p$ spaces since I was never formally intrduced to them through courses. I wanted to know if anyone could offer me a proof or give ...
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### Isometries on the Banch Space M([0,1]) of regular Borel Measures

I'm trying to define an isometric isomorphism $T:M([0,1])\to M([0,1])$ that is not weak-star continuous (by $M([0,1])$ I mean the Banach space of regular Borel measures). How I can build one? One ...
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### A problem of maximization in Banach spaces

Let $X$ be a Banach space and $K\subset X$ compact. Let $C(K)$ be the set of continuous function in $K$ and $\mu\in (C(K))^\star$ a non-negative measure. Assume that $f:K\to X^\star$ is a continuous ...
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### Banach-Mazur distance

I am stuck upon the following problem. Consider the Banach-Mazur distance for $X$ ,$Y$ normed isomorphic vector spaces $$d(X,Y) = \inf \{ \| T \| \| T^{-1} \| : T \in GL(X,Y) \}$$ I would like to ...
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### Schauder basis for a separable Banach space

It is known that if a Banach space $X$ has a Schauder basis, then $X$ is separable. On the other hand P. Enflo showed that there exist a separable Banach space without Schauder basis. If $X$ is a ...
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### Norm on a Banach space

Let $\left( X, \| \cdot \|\right)$ be a Banach space over some field $\mathbb{K}$. Let $x$ be fixed in $X$ such that $\|x\| \le 1$. If $x_0$ is any point in $X$ , I need to show that there ...
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### Banach fixed point theorem and inverse function

Let $U$ and $V$ be the open subsets in $\mathbb{R}^n$, $x\in U$ and $f:U\rightarrow V$ is a smooth function. There is an inverse function theorem which states that if the Jacobian determinant at $x$ ...
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### Topology induced bycone metric

Is cone metric define atopology as same as the topology define by ametric? I have tried to prove it by theorems that joined them
### Proving norm equivalence in $W^{1-1/p,p}(\Omega)$
Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...