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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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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 exists $\alpha $ in $\mathbb{K}$ such that $\| x+ \alpha…
CeCe
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Projectors on Banach spaces for unions and intersections

It is well known that not every closed subspace of a Banach space admits an associated idempotent operator. Now suppose that we have a filtration $(S_i)_{i\in\mathbb{Z}}$ of closed subsets of a Banach space $B$, such that for all $i$, $S_i\subseteq…
geodude
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Equivalent bases in Banach spaces definition

Two basic sequences $(x_n)$ and $(y_n)_n$ in Banach spaces $X$ and $Y$ are said to be equivalent if there is an isomorphism $T$ mapping the closed linear span of $(x_n)_n$ onto the closed linear span of $(y_n)_n$ mapping $Tx_n=y_n$. Is the idea that…
user124910
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Distance between two points in a totally bounded set in a Banach space is finite?

I am doing the proof that every totally bounded set in a Banach space is bounded. The fact that a set is totally bounded means that for every $\epsilon >0$ there are points $x_1,...,x_n$ such that $B_\epsilon (x_j)$, $j = 1,...,n $ is a covering of…
mikasa
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What is the norm in the definition of the projective norm?

The projective norm is definied as: $\pi(u)=\inf\left\{\sum_{n=1}^{\infty} \|x_n\|\|y_n\|:\sum_{n=1}^{\infty} \|x_n\|\|y_n\|<\infty,\, u = \sum_{n=1}^{\infty} x_n \otimes y_n \right\}$, what are the norms $ \|x_n\|$ and $\|y_n\|$?
asterix
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Prove that $\mathbb{C}$ with the absolute value norm is a Banach space.

Prove that $\mathbb{C}$ with the absolute value norm is a Banach space. In this link, we can read a proof for the real case. Is the proof for $\mathbb{C}$ the same?
Mark
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Range of operator being a limit

Let $X$ be a Banach space and let $T_n\colon X\to X$ be a family of bounded operators convergent to some operator $T\colon X\to X$. Is it true that $T(X)\subseteq \sum_{n=1}^\infty T_n(X)$? I mean by $\sum_{n=1}^\infty V_n$ the set of all (finite)…
dziobak
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Infinite product of reflexives Banach spaces

If $X_1$ and $X_2$ are reflexive Banach spaces and $p\in[1,\infty]$, then the product $X_1\times X_2$ with the norm $\|(u,v)\|=(\|u\|^p+\|v\|^p)^{\frac1p}$ is also a reflexive Banach space (all these norms are equivalent; the $\infty$-norm is simply…
PHL
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Compute norm in a Banach space

Let $I = [0, 1] \subset \mathbb{R}$ and the scalar field is $\mathbb{R}$. For a Banach space $C(I)$, let $\Lambda(f)=\int_{0}^{1}\left(9 t^{4}-18 t^{3}+11 t^{2}-2 t\right) f(t) d t$ I would like to calculate $\|\Lambda\|$. I used the fact that f is…
alryosha
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Does this notion of project in Banach spaces exist?

Let $X$ be a Banach space and let $W$ be a linear subspace. Consider some $v$ in $X$ and let the projection of $v$ onto $W$ be defined as the $u$ in $W$ st. $\forall w\in W,\|v-u-w\|=\|v-u+w\|$. If there is a unique $u$ for which this holds, call…
Mathew
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finding a unit vector of maximum distance from a subspace in a banach space

Let $W$ be a strict subspace of a Banach space $X$ and let the distance between a vector $v$ and the subspace $W$ be defined as $d(v,W)=\inf_{w\in W}|w-v|$. If $X$ is a Hilbert space, then we can always find a unit vector $v$ st. $d(W,v)=1$. Does…
Mathew
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Limit of average of bounded linear functionals on finite dimensional normed vector space

Suppose I have a finite dimensional normed vector space $V$, and a sequence of $f_i$ in $V^*$ with norm all equal to 1. I want to know when does $\lim_{n->\infty} \frac{1}{n}\sum_{i=1}^{n} f_i$ converges. Now because $||f_i|| = 1$ for all $i$, I…
Kerwin Yi
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Banach subsequence converges

Show that every sequence in a Banach space such that $\{x_n\} \rightarrow 0$ has a subsequence $\{x_{n_p}\}$ such that $\sum_{p=1}^{\infty} x_{n_p} $ converges by showing $S_N = \sum_{p=1}^{N} x_{n_p}$ is a Cauchy sequence. The convergent to the…
user79018
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Prove that if $E \simeq E'$, $E \bigoplus F \simeq E' \bigoplus F'$ then $F \simeq F'$.

All the spaces in this question are Banach Spaces, and "$\simeq$" means they are linear homeomorphic. The question is to prove the following: Prove that if $E \simeq E'$, $E \bigoplus F \simeq E' \bigoplus F'$ then $F \simeq F'$. Any hints on how to…
Ulivai
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Completeness of $L^\infty$

I found a simple proof for the completeness of $L^\infty$, based on the fact that a normed vector space is complete if $$\sum_n ||f_n||<\infty \implies \exists y=\sum_n f_n$$ I am not sure that the last passages work. Let $f_n$ be a sequence in…
Davide Maran
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