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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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Let $X$ be a Banach space, and $B_1\supseteq B_2 \supseteq\cdots $ . Show that $\bigcap\limits_{i=1}^\infty B_i\neq\emptyset$

Let $X$ be a Banach space, and $B_1\supseteq B_2 \supseteq \cdots $ a sequence of closed balls with radius $r_i$ and center $x_i$. Show that $$\bigcap_{i=1}^\infty B_i\neq\emptyset$$ I proved that $r_i\leq r_j$ for $B_i\subseteq B_j$ and…
José
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Complemented subspaces of $c_0(\Gamma)$

Let $\Gamma$ be an infinite set. By $c_0(\Gamma)$ we denote the Banach space of all functions $f: \Gamma \to \mathbb{R}$ such that, for all $\varepsilon > 0$, the set $\{ \gamma \in \Gamma : |f(\gamma)| \geq \varepsilon\}$ is finite, equipped with…
Vinícius Morelli
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Showing $(\ker(T))^{\circ} = \operatorname{Im}(T')$ in Banach space

For $T: X \to Y$ with $T$ having finite dimensional image I'm trying to show: $$(\ker(T))^{\circ} = \operatorname{Im}(T')$$ Where $T'$ is the dual operator. I've shown that if we take an element in $\operatorname{Im}(T')$ this is zero on the kernel…
Wooster
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Banach space is sum of $ker f$ and $X \ ker(f)$.

I'm trying to show that if $f$ is an element of the dual space $X^*$ of a Banach space, $X$, and $x_0 \in X-ker(f)$, then every element in $X$ can be expressed as $x = \lambda x_0 + y$ with $y \in ker(f)$. I feel like this should be trivial to…
Wooster
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Power Series for a Banach Contractive Mapping function

For each continuous function $f \in C[0,1]$ define the continuous function $T(f)(x)$ by $$T(f)(x)=x^2 + \int_{0}^{x}tf(t)dt$$ for each $x\in C[0,1]$. I'm trying to find the power series to represent function f(x) and I am not sure I/m on the right…
Maggy Wood
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Is Limit and Norm interchangable in Banach Spaces

Suppose $X$ is a Banach space, and $\{x_n \}\subset X$. Does it then hold that $\lim \|y-x_n\|=\|y-\lim x_n \| $ ?
user112110
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Question about Hamel's basis.

I'd like to prove that if I consider $X$ Banach space with $\dim X =\infty$, then $X$ can't have a countable Hamel's basis. Someone can give me a hint? Even a counterexample is enough. I think this can be linked with Baire's theorem about Banach's…
rusca91
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Quotient of $\ell_1$ by space of finite sequences

Consider $\Phi$ to be the space of sequences that have finitely many non-zero terms. The space is not closed in $\ell_1$, therefore $\ell_1/\Phi$ with the quotient topology is not Hausdorff, and so it cannot be metrizable. However, does there exist…
Ivan
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Sequence of the linear operators on Banach spaces

Let $X$ and $Y$ be Banah spaces and let $(T_n)$ be a sequence of bounded linear operator from $X$ to $Y$. I need to prove that following statements are equivalent: (a) Sequence $(||T_n||)$ is bounded (b) Sequence $(||T_n(x)||)$ is bounded for each…
Laki888
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Bounded sequence in vector valued space

I am working with a space of the form $E=L^{\infty}(0,\infty;X)$, where $X$ is a non reflexive Banach space. If a sequence $(x_n)$ is bounded in $E$, how to extract a weakly converging subsequence? I think that this is not always possible for an…
elmas
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retraction of closed unit ball to its boundary

Let $X$ be a Banach space, $\overline{}$ be its closed unit ball and $$ its unit sphere. A retraction from $\overline{}$ to $$ is a continuous map $r:\overline{}→$ such that $\left.r\right|_$ is the identity map on $$. When $X$ has finite dimension,…
hbghlyj
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Prove that $(B(A),\|\cdot\|_\infty)$ is a Banach Space

I need to prove that $(B(A),\|\cdot\|_\infty)$ is a Banach Space, where $B(A)$ is the set of bounded functios from $ A $ to $\mathbb R$. I have seen the proof but I don't understand it. This is what I have: First, we take ${f_n}$ a Cauchy sucession.…
Elena
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Clarification for the definition of compact operators

One of the definitions of compact operators is: For any bounded sequence $(x_{n})_{n\in \mathbb {N} } \in X$, the sequence $(Tx_{n})_{n\in \mathbb {N} }$ contains a converging subsequence. Does this mean that the limit is in $Img(T)$? My problem is…
Anon
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Lifting of Reflexivity

I am looking for a proof (the simple the better) of the Theorem stating that $L_p(E)$ is reflexive if and only if $E$ is reflexive. Here is $E$ a Banach space and $L_p(E)$ is the Lebesgue-Bochner space of strongly measurable functions with their p…
PaulDalex
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Banach subspace of Banach space

How to prove that subspace of Banach space is also Banach space? I believe that's only true if subspace is closed. However, I don't understand why we can't take non-closed subspace. Is there any counterexample of non-closed subspaces that will not…
NixoN
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