# Questions tagged [banach-spaces]

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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### Reference on separability of $c_0(X)$, where $X$ is a separable Banch space

Let $(X,\|\cdot\|)$ be a separable Banach space. By $c_0(X)$ I mean the space $\{(x_n)_n\subset X:\, \|x_n\|\to0\}$. I think it os well known that $c_0(X)$ is a separable Banach space endowed with ...
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### Proving $\| g^{-1}-f^{-1}\|\leq 2\|{f^{-1}}\|^2\|g-f\|$ [duplicate]

Let $f,g\in L(X,Y)$ where $X,Y$ are Banach spaces and let $f^{-1},g^{-1}\in L(Y,X)$. How to prove that $$\| g^{-1}-f^{-1}\|\leq 2\|{f^{-1}}\|^2\|g-f\|$$ if $\|g-f\|\leq\frac{1}{2\| f^{-1}\| }$?
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### The Relationship between Reflexive Space, Separable Space and Compactness

I'm currently studing functional analysis. It bothers me a lot about the relationship between reflexive space, separable space, and compactness of unit ball (in different space, in different topology)....
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### Showing incompleteness of metric space by specifying non-convergent Cauchy sequence

I have been looking at the space of continuous functions over a compact interval $C([0,2])$ equiped with the the integral norm of absolut values $\| \cdot \|_1$. I read a counterexample that showed ...
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### A metric for weak* convergence on bounded sets

Let $X$ be a separable Banach space and let $(x_n)_{n \in \mathbb{N}}$ be a fixed sequence such that $x_n \neq 0$ for all $n \in \mathbb{N}$ and $\lbrace x_n: n\in \mathbb{N}\rbrace$ is dense in $X$. ...
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### Logarithm of the norm of a Banach space-valued holomorphic function is subharmonic?

Given a holomorphic function $f$ in some open subset $G\subset\mathbb{C}$, it is well-known that the real-valued function $z\in G\mapsto\log|f(z)|\in\mathbb{R}$ is a \emph{subharmonic} function, i.e., ...
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### About bounded operator in normed space

Suppose that $T$ is a linear operator from a normed space $X$ into a normed space $Y$ such that $\sum_{n=1}^{\infty} T (x_{n})$ is a convergent series in $Y$ whenever $\sum_{n=1}^{\infty} x_{n}$ is ...
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### Space of Lipschitz functions (with some conditions) form a Banach space

Let $X=[0,1]$ and \begin{equation*} L = \lbrace f:X\rightarrow \mathbb{R}:f \ \text{is Lipschitz}, f(0)=0 \ \text{ and }\ f(1)=1\rbrace \end{equation*} with norm \begin{equation*} \|f\|=\sup_{x\neq ...
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### What is the geometrical meaning of the space?

Consider the Banach space of all lipschitz functions on $X$ such that for each $f$ in the space, \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\...
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### How to prove that whether it is a Banach space or not?

We consider the Banach space of all continuous functions on $X$ such that for each $f$ in the space, \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\...
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### Show that: for all $x^*\in D$: $\lim_m\frac{1}{m}\sum_{n=1}^{m}{\sup_{x\in A_n^k}{\langle x^*, x\rangle}}=\beta_{x^*}^k$

Let $X$ be a separable Banach space. $X^*$ will denote the dual of $X$ and $\langle,\rangle$ the usual duality. Put: $$\mathcal{P}_{c}=\{C\subset X:C\text{ is nonempty closed convex subset of }X\}$$ ...
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### Open cover of $A\subseteq\mathbb{R}$ such that it does not have a finite subcover.

If $A\subseteq\mathbb{R}$ is not closed. How can I build an open cover of $A$ such that it does not have a finite subcover?
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### Spectrum of restriction to invariant subspace

Let $H$ be a separable Hilbert space and let $\mathcal{B}(H)$ denote the algebra of linear bounded operators on $H$. Let $T \in \mathcal{B}(H)$ and let $M$ be a non-trivial closed invariant subspace ...
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### Textbook recommendations for weak topology

This is not a question regarding a specific mathematics problem, rather I am looking for some good texts that go into detail on the weak topology. My exposure to weak topologies is via Banach spaces ...
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### Distance to the kernel of a functional in Hilbert space

I need to prove that in Hilbert space $H$ the distance between point $z$ and the kernel $L$ of a linear functional $f$ is $d(z,L)=\frac{|f(z)|}{\|f\|}$. I know a rather sophisticated proof for Banach ...
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### Extreme point of $B_A = \{a\in A: \|a\| \leq 1\}$ in a Banach algebra $A$

I've send this question a little while ago now I edit it Let $A$ be a Banach algebra with identity $e_A$ prove that $e_A$ is an extreme point of $B_A = \{a\in A: \|a\| \leq 1\}$ Recall ...
### Algebraic dimension of the dual of $\ell^{\infty}$
It is shown that the algebraic dimension of an infinite dimensional Banach space is equal to it's cardinality. Using this fact, can we say that the algebraic dimension of $(\ell^{\infty}(X))^*$is ...
### Prove $e_A$ is an Extreme point of $B_A = \{a\in A: \|a\| \leq 1\}$ in a Banach algebra $A$
Let $A$ be a Banach algebra with identity $e_A$ prove that $e_A$ is an extreme point of $B_A = \{a\in A: \|a\| \leq 1\}$ Recall that $x$ is an extreme point of $B_A$ if \$x = \frac{1}{2}(x_1+...