Mathematics Stack Exchange
Mathematics Stack Exchange

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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What is $\lVert c \rVert_{X}$ where $c$ is constant?

A thought just occurred to me, in a Banach space $X$, what is $$\lVert c \rVert_{X}=c\lVert \text{Id} \rVert_{X}$$ where $c$ is a constant? Is it even defined?
matt.w
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Prove that the set is a banach space

I have the following prove which I don't understand from a certain point: Let be $ \mathcal{B}(X,\mathbb{R}) $ the set of bounded functions and $ \|\cdot\|_{\infty} $ the uniform norm. Then $ (\mathcal{B}(X,\mathbb{R}),\|\cdot\|_{\infty}) $ is a…
hallo007
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If $x_n\in l^1(\mathbb{K})$ and $x_n\rightarrow 0$ in $X_p$, then $x_n\rightarrow 0$ in $X_1$?

Is it true that if $x_n\in l^1(\mathbb{K})$ and $x_n\rightarrow 0$ in $X_p$, then $x_n\rightarrow 0$ in $X_1$? Where $X_1= (l^1(\mathbb{K}),\|.\|_1)$ and $X_p=(l^1(\mathbb{K}),\|.\|_p)$ I know that $l^1(\mathbb{K})$ is subspace of $l^p(\mathbb{K})$.
kida
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To understand the distance function in (X, || . || ) × ( X, ||. || )

A very silly doubt but I don't know what I am thinking. See , book introduced the the notion of inner product and then also defined how we can get norm from it. Then book is saying <.,.> : (X, || . || ) × ( X, ||. || ) $\to$ F(scalar field) is…
ogirkar
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If Banach space $C(K)$ is decomposable (non trivial) then $C(K)$ is isomorphic to some of the sumands subspaces

In Beauzamy (Banach spaces) book appears this statement without proof: "if $X\oplus Y$ is isomorphic to Banach space $C(K)$ then either $X$ or $Y$ is isomorphic to $C(K)$'' where $X$, $Y$ are Banach spaces and $C(K)$ is the Banach space …
Aligomez
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Approximating a Banach space valued function by sums of continuous functions

I am trying to prove the following exercise, which is a part of a project type homework problem. Please give hints and suggestions, and discuss this problem. Let $(T,d)$ be a compact metric space, and let $\mathcal{X}$ be a Banach space over the…
user82261
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Is this subspace isomorphic to $\ell^1$?

Let $X$ be a Banach space. Suppose there exists a sequence $(x_n)$ in $X$ such that for all finite $A\subseteq\mathbb{N}$ we have that $\|\sum_{n\in A}x_n\|$ equals the number of elements in $A$. Does this imply that the subspace spanned by…
Carucel
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Question about $\ell^{p}$ spaces

I am quite new to the subject of sequence spaces. I got a few doubts (hope they are not silly). While reading about $\ell^{p}$ spaces, I read that these spaces equipt with the $p$-norm form normed linear spaces. My question is every time I am…
ogirkar
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Map which is bijective and continuous, but not open

I have bigger problems with these 2 questions. a) Let $\mathbf{X}$ be a Banach space and let $\mathbf{S:X→R}$ be linear and not bounded. Let $\mathbf{Y = graph(S) ⊂ X×R}$. Show that the map $\mathbf{T: Y → X}$ given by $\mathbf{T((x,Sx)) = x}$ is…
user574356
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Understanding the $\sigma(X^{\ast\ast},X^\ast)$ topology

I'm trying to prove that $X$, a Banach space, is dense in its bidual $X^{\ast \ast}$ with respect to the $\sigma(X^{\ast\ast},X^\ast)$ topology. I'd like some help. In particular, I know that the $\sigma(X^{\ast\ast},X^\ast)$ topology is the…
Drew Brady
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Showing that the limit exists for a function in a Banach space

Let E be a normed space and F a Banach space. Suppose we have A ⊂ E, $x_o$ ∈ Acc(A) and f : A → F , a function uniformly continuous. Show that lim x→x0 of f(x) exists.
chadi
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Extension of Linear Operatos

Possible Duplicate: Do continuous linear functions between Banach spaces extend? Is there an example of a pair of Banach spaces $X$ and $Y$, a subspace $E\subseteq X$ and a bounded linear operator $T:E\rightarrow Y$ (with the norm on $E$ induced…
Luiz Gustavo Cordeiro
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Does any closed subset in a Banach space have (at least one) point that has a minimum norm?

Does any closed subset in a Banach space have (at least one) point that has a minimum norm? I think this statement is obviously true, but how do I prove its correctness?
xzhu
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Representation of a bounded linear operator in the form of a matrix

Let $X$ be a Banach space and $M$ be a subspace of $X$. Then we know $X= M\oplus N$ where $N$ is a subspace of $X$. Let $T:X \to X$ be a bounded linear operator. I want to prove that $ T= \begin{pmatrix} A&B\\ C&D\\ \end{pmatrix} $ where $A:M…
Manu Rohilla
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Generalisation of Riemann theorem

We have, from real analysis, that: Let $a_n$ be an array in $\mathbb{R}$, such that $\sum_{n=0}^{+\infty} |a_n|$ converges, then, for any bijection $\phi:\mathbb{N} \to \mathbb{N}$ it holds $\sum_{n=1}^{+\infty} a_{\phi(n)} = \sum_{n=1}^{+\infty}…
nikola
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