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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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Unit ball in a separable Banach Space

Suppose that $X$ is a separable Banach space. Then there exists a sequence $\{ x_n: n \in \mathbb{N} \}$ such that the sequence is dense in a unit ball $B_X$. Question: Whenever we talk about separable Banach space, I notice that we always have a…
Idonknow
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complemented subspaces of $L_{p}$ spaces (Question posed incorrectly earlier)

This question was asked incorrectly originally in such a way that it probably made no sense. Fixed version: I know that $L_{p}[0,1]$ has $\ell_{2}$ as a complemented subspace, and I'm wondering if this can be generalized to include $\ell_{q}$ for…
Kevin
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Second dual of a Grothendieck space

The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{****}$, $\dots$ are Grothendieck spaces. (See, e.g., this paper for the proof of Grothendieck property of these…
ald.li
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Banach space "over $\mathbb Q$"

My question: can we define Banach space over the field $\mathbb Q$ instead of over the usual $\mathbb R$ and $\mathbb C$? Will $\mathbb R$ be a Banach space over $\mathbb Q$ if it can be done? The reason why I am asking this is because I have read…
Rajat
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Sum of the sets in a Banach space

I have the following question. Let $M$ be the closed unit ball with center in $0$ and radius $1$ in a Banach space $X$, so $M=B_1(0)$. Let $f,g$ be linear und bounded $X\to X$. Is it true, that $\overline{ (f+g)(M) }\subset…
Logic_Problem_42
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$\mathbb N$ a Banach space?

Is $\mathbb N$ a Banach space with the norm $|x-y|$ from $\mathbb R$? I think is Banach space because there is no convergent sequence that is not constant after some $N$. Then all limit points are in the space. But I am not sure.
blue
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Urysohn-like theorem in Banach spaces

I have a (separable) Banach space $E$ and two closed disjoint sets $F$, $G$ in $E$. Now I wish to prove the existence of a $C^2$-function (Fréchet differentiable) $f:E \to \mathbf R$ that is $1$ on $F$ and $0$ on $G$. Does someone have a reference…
JT_NL
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Bounded linear operator on Banach space

For all $f\in C[0,1]$ and $x\in[0,1]$ we define bounded linear operator $$(Tf)(x)=f(0)+\int_0^x f(t)dt,$$ with norm $||T||=2$ (right?). I have already determined its image $imT:=\{g\in C^1[0,1]| g'(0)=g(0)\} $. Question: Is $imT$ Banach space with…
fasdgr
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Linear mapping between Banach space and its dual

This problem has been giving me some troubles. Does anyone have any ideas on how to go about proving this? Let X be a Banach space and let $$\Phi: X \rightarrow X^*$$ be a linear mapping such that for each $x\in X$ we have $\Phi(x)(x)=0.$ Then: For…
fasdgr
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A question about why a space under a certain norm is complete

A theorem I am reading (about the existence and uniqueness of solutions to Sturm-Liouville intial-value problems) defines a space $B$ consisting of the continuous functions defined on a closed real interval $[a,b]$ and assuming values given by…
Josef K.
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Bi-dual norm in a Banach space

Suppose $X$ is a Banach space, $X^*$ its dual and $X^{**}$ the dual of the dual. Then, for $x\in X$, we can define $F_x \in X^{**}$ as $$ F_x(\phi) = \phi(x) $$ for every $\phi \in X^*$. Then, it is clear that $\|F_x\| \leq \| x \|$, but does the…
nan
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What does well-isomorphic mean?

What I'm currently reading discusses the notion of spaces being well-isomorphic, specifically a Banach space containing well-isomorphic copies of $\ell_1^n$ for every $n\in\mathbb{N}$. The author doesn't define this, so I gather it's an elementary…
user348871
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Sequences - Convolution

We got the following setting: Let $A$ the set of all complex sequences $x=(x_n)_{n \in \mathbb N_0}$ with \begin{align*} \Vert x \Vert := \sum_{n = 0}^\infty \vert x_n \vert e^{-n^2} < \infty. \end{align*} Consider the convolution $(xy)_n = \sum_{k…
Yaddle
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Classification of subsymmetric basic sequences

I just encountered the following statement: any subsymmetric basic sequence is either weakly null or equivalent to the unit vector basis of $\ell_1$. I see the case in which it is equivalent to the unit vector basis of $\ell_1$. However, I cannot…
ragrigg
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An application of Riesz' Lemma

How does one prove using Riesz' Lemma that an infinite dimensional subspace $Y$ of a Banach space $X$ contains a sequence $\{x_n:n\in \mathbb{N}\}$ in the unit ball of $Y$ such that $n \neq m$ implies that $\|x_n−x_m\|>1/2$?
johnathan
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