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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.

A Banach space, named after Stefan Banach (1892–1945) is a complete normed vector space: a (real or complex) vector space equipped with a norm such that every Cauchy sequence converges. For instance, $\mathbb{R}^n$ and $\mathbb{C}^n$, equipped with the usual norm (or, for that matter, any norm) is a Banach space. Another example is the space $\ell^1$ of all absolutely convergent series of real or complex numbers, equipped with the norm $\left\|\sum_{n=0}^\infty x_n\right\|=\sum_{n=0}^\infty|x_n|$.

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Non-closed subspace of a Banach space

Let $V$ be a Banach space. Can you give me an example of a subspace $W\subset V$ (sub-vectorspace) that is not closed? Can't find an example of that yet. Thanks!
Sh4pe
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Is product of Banach spaces a Banach space?

If $X$ is a Banach space, then I want to know if $X\times X$ is also Banach. What is the norm of that space? So for example, we know $C^k(\Omega)$ is Banach and I have a vector $v = (u_1, u_2)$ where $u_i \in C^k(\Omega)$ and I want to use…
blahb
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When can a real Banach space be made into a complex Banach space?

Suppose I have a real vector space $V$ and I would like to extend the scalar multiplication in such a way that I obtain a complex vector space. It is not difficult to see that doing so is equivalent to fixing some linear map $U : V \to V$ satisfying…
Mike F
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Equivalent definitions of Injective Banach Spaces

A Banach space $X$ is said to be injective if for all Banach spaces $W,Z$ with $W\subset Z$, and operators $T\in B(W,X)$, $T$ can be extended to all of $Z$ with the same norm. Equivalently, $X$ is injective if it is complemented by a norm $1$…
roo
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Bounded Linear Mappings of Banach Spaces

This problem has been giving me some troubles. Does anyone have any ideas on how to go about proving this? Let $X$ and $Y$ be Banach spaces. If $T: X \to Y$ is a linear map such that $f \circ T \in X^*$ for every $f \in Y^*$, then $T$ is…
breeden
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How to define Continuous $f: B\rightarrow B$ without fixed points?

Let $X$ be a infinite dimensional Banach space and $B=\{x\in X:\ \|x\|\leq 1\}$. How to define a continuous $f:B\rightarrow B$ without fixed points? Edit 1: I have changed the question a litlle, because if $f$ is linear then $f(0)=0$.
Tomás
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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 $||(a_{\lambda})||=\sup_{\lambda}||a_{\lambda}||<\infty$. I am trying…
cyc
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Homeomorphisms on X and automorphisms on C(X)

Let $ X $ be a compact Hausdorff space. Let $ \psi $ be a homeomorphism on $ X $. Let $ \text{Aut}(C(X)) $ be the group of automorphisms of $ C(X) $, and $ \text{Homeo}(X) $ be the group of homeomorphisms on $ X $. Show that the mapping $ \Psi:…
john
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Basis properties of the polynomial system in the space of continuous functions

Consider $\{1, t, t^2, t^3, \ldots\}$ as a subset of $C\big( [0, 1]\big)$. Clearly this system is linearly independent. Also, it is a complete system (meaning that its closed linear hull is the whole space $C\big( [0, 1]\big)$ ) but it is not a…
Giuseppe Negro
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Subspace isomorphic to a complemented subspace

I'll begin by writing down the definitions I'm using, to avoid confusion. Let $X$ be a Banach space and let $Y$ be a subspace of $X$. We say that $Y$ is complemented in $X$ if there exists a linear continuous operator $P : X \to Y$ such that Im$(P)…
Vinícius Morelli
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Exercise in Folland on extending a closed subspace of a Banach space

Folland gives the following problem on page $159$ of his book Real Analysis: Let $\mathcal X$ be a normed space. If $\mathcal M$ is a closed subspace and $x\in \mathcal X \setminus \mathcal M$ then $\mathcal M + \mathbb C x$ is closed. The only…
Potato
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reflexive Banach space

Let $X$ be a Banach space and $B_X$ be the unit ball. Suppose that for each $\lbrace C_n\rbrace_{n=1}^\infty\subset B_X$ satisfying $C_n$ are closed convex and $C_n\supset C_{n+1}$ has a nonempty intersection. Is it true that $X$ is reflexive?
Jack Smith
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A certain element which makes functionals positive

Suppose $X$ is a possibly non-separable Banach space and let $X^*$ be its dual. Also, let $(f_n)_{n=1}^\infty$ be a sequence of [EDIT: linearly independent] norm-one functionals in $X^*$. Does there exist an element $x\in X$ such that $f_n(x) \in…
user60253
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$\ell_\infty^* $ has the Dunford--Pettis property

I'm trying to prove that $\ell_\infty^*$ has the Dunford–-Pettis property. It's enough to show that $\ell_\infty$ does not contain a copy of $\ell_1$ … but I'm having some trouble doing that. Can anyone help me ? Here's my approach. I found this…
Rafael
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Separable, reflexive space that is almost isometrically universal for the class of finite dimensional normed spaces.

My question is whether there is a separable, reflexive Banach space $X$ s.t. for any finite dimensional normed space $E$ and ${\epsilon>}0$, there is a linear isomorphism $T$ from $E$ to its image in $X$ s.t $||T|| ||T^{-1}||<1+\epsilon$? For…
Mathmo
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