A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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Exercise of series in a Banach Space

A vector basis of a vector space $E$ is a family $(a_{\lambda})_{\lambda\in L}$ such that any element of $E$ can be written in a unique way as a linear combination of a finite number of $a_{\lambda}$, ...
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468 views

Does there exist such a closed subspace of normed linear space

let $(X,|| || )$ be a norm linear space. And $M$ be a closed subspace of norm linear space .does there exist a closed subspace $N$ such that $X=M \oplus N $ . I know such an subspace $N$ exist .but i ...
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1k views

How is the weak-star topology useful?

Today I learnt something about the weak-star topology, but I don't know what the use of weak-star topology is. I hope someone can tell me what we can do with the weak-star topology. Thanks in advance! ...
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1answer
334 views

Every quotient of a reflexive space is reflexive

How do you prove the following? If $\mathcal{X}$ is reflexive and $M \leq \mathcal{X} \rightarrow \mathcal{X}/M$ is reflexive There is no assumption that $\mathcal{X}$ is a Banach space.
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Closed operator

I've got a very straightforward question : if $T : B \rightarrow B$ is a linear continuous operator and $B$ is a Banach space, is $T$ a closed operator? This is obviously true in finite dimension, ...
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Why are $C(K)$-spaces $\mathcal{L}^{\infty}$-spaces?

Let me first recall the definition of an $\mathcal{L}^{\infty}$-space: A Banach space $X$ is called an $\mathcal{L}^{\infty}$-space if there is a net $(X_{\lambda})_{\lambda \in \Lambda}$ (directed ...
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75 views

function in $L^1\setminus L^2$

I'm looking for an example of a function which belongs to the Banach space $L^1$ (i.e $\int|f|< \infty$) but is not in $L^2$ (so $\int|f|^2$ is unbounded). Does anyone know such a function?
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weak* separable question

(In another question Nate Eldredge said I should ask this.) Let $X$ be a Banach space, $X^\ast$ the dual space, and $B_{X^\ast}$ the unit ball of $X^\ast$. In the weak* topology for $X^\ast$, does ...
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631 views

Example of an unbounded operator

Can somebody give me an easy example of a linear operator which maps $L^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$ and $L^\infty(\mathbb{R}^n)$ to $L^\infty(\mathbb{R}^n)$ (but not boundedly) but does ...
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1answer
228 views

Continuity of bilinear form

Let $X$ be a Banach space and $b(\cdot,\cdot):[0,C] \times X \to \mathbb{R}$ be a form that is continuous wrt. the first argument, linear wrt. the second argument and satisfies $$b(t,x) \leq K\lVert x ...
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1answer
692 views

If $X^\ast $ is separable $\Longrightarrow$ $S_{X^\ast}$ is also separable

Let $X$ be a Banach space such that $X$* (Dual space of $X$) is separable How can we prove that $S_{X^\ast}$ (Unit sphere of $X$*) is also separable Any hints would be appreciated.
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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 ...
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1answer
69 views

$\lVert A\lVert<1$ implies $1-A$ invertible only true in complete spaces?

It is a well known fact that if in a Banach space $X$ a bounded linear operator $A:X\to X$ satisfies $\lVert A\lVert<1$, then $1-A$ has a bounded inverse. I was wondering wether completeness ...
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1answer
68 views

Banach Space Condition help

Is $X$ a Banach space, $b\in x$ a vector and $A\in L\left ( x,x \right ) $ an application that checks the condition $\left |A ^{n} \right |< 1$, for all $n\in \mathbb{N}$. (a) show that the ...
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80 views

Is the adjoint operation WOT-WOT continuous?

This is a well-known property of the Hilbert-space adjoint operator that it is WOT continuous. Is a similar true for Banach spaces? That is, for a given Banach space $X$ is the operation ...
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97 views

An orbit of a group action and the implicit function theorem

Suppose that a Lie group $G$ acts smoothly on a manifold $X$. We can easily prove the following theorem by using the constant rank theorem, which is a stronger theorem than the implicit function ...
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299 views

$C^{1}[a,b]$ equipped with the norm given by $\lVert x\rVert _{\infty} = \sup_{t\in [0, 1]} \lvert x(t) \rvert$ is an incomplete normed space.

I have to show that the real linear space $C^{1}[a,b]$ of all continuously differential functions defined on $[0, 1]$ equipped with the norm given by $\lVert x\rVert _{\infty} = \sup_{t\in [0, 1]} ...
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1answer
2k views

$C([0, 1])$ is not complete space with respect to norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$

Consider $C([0, 1])$, the linear space of continuous complex-valued functions on the interval $[0, 1]$, with the norm $\displaystyle\lVert f\rVert_1 = \int_0^1 \lvert f(x)\rvert \,dx$. I have to ...
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1answer
279 views

Can we visualize the closed balls for the space $l^{\infty}$ equipped with the $\sup$ norm

The closed unit balls for the $l^{p}$ in $\mathbb{R}^2$ look like I want to know could we also visualize the closed balls for the space $l^{\infty}$ equipped with the $\sup$ norm . Thanks
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Geometry of the space $C[a, b]$ respect to the norm $\lvert \lvert x \rvert\rvert_{\infty} = \max_{t\in [a.b]}\lvert x(t)\rvert$.

I have studied that the space $C[a, b]$ of all scalar-valued (real or complex) continuous functions defined on [a, b] is a Banach space with respect to the norm $\lvert \lvert x \rvert\rvert_{\infty} ...
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How to prove that sequence spaces $l^{p}, l^{\infty}$ and function space $C[a. b]$ are of infinite dimension

I am studying about the sequence space $l^{p}, l^{\infty}$ and function space $C[a. b]$. It is mentioned in the book that all of these spaces are of infinite dimension. I want to prove that these ...
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233 views

Compact subsets of function spaces, geometry

The subset is called compact when every open cover contains a finite subcover. In Euclidean spaces, it is easy to visualize this by imagining some open ball that contains this set, thinking about the ...
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1answer
65 views

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} ...
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805 views

Show reflexive normed vector space is a Banach space

$X$ is a normed vector space. Assume $X$ is reflexive, then $X$ must be a Banach space. I guess we only need to show any Cauchy sequence is convergent in $X$.
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83 views

Why this space is not a complete space with this norm

Show that the space $C_0(\mathbb{R})$ of all the real continuous functions $f:\mathbb{R} \to \mathbb{R}$ with compact support is not a complete space with the norm $||f||= \sup_{t∈ \mathbb{R}}|f(t)|$. ...
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1answer
170 views

Is a Banach Space

Show that the vector space, $P_n$, of all the real polynomial functions of degree less than n, is a Banach Space for any norm define. I think if I prove that $P_{n}$ is a Banach Space with the norm ...
2
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1answer
170 views

Showing atomic $H^{1,p}$ is a Banach Space

Consider $1<p\leq\infty$ and let $H^{1,p}=H^{1,p}(\mathbb{R}^n)$ be an atomic Hardy space, that is $H^{1,p}\subset L^1(\mathbb{R}^n)$, and $f\in H^{1,p}$ if there exists a sequence of $p$-atoms ...
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360 views

Proof of non-strictly convexity of $l_1$ and $l_{\infty}$

Given the Banach space $l_p = \left\{f\in R^n : \|f\| \leq \infty\right\}$ for $1\leq p \leq \infty$, we define the following norms $\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty} ...
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Norm inequalities in a reflexive space

I am reading an article about reflexive spaces, with a specific example. The article mentions inequalities that I haven't been able to get around to. Here's the setup. The space $X = (\prod_n ...
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1answer
205 views

an example to show separability of a Banach space does not imply separability of the dual space

$X$ is a Banach space and it is separable, is there any simple counterexample to show the dual space $X^\ast$ is not separable?
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Does every Banach space admit a continous injection to a non-closed subspace of another Banach space?

Let $(V, \lVert\,\rVert)$ be a Banach space. I want to produce a non-complete norm $\lVert\,\rVert'$ on it such that $\lVert v\rVert' \leq \lVert v\rVert$ for all $v$ in $V$. Given a continuous ...
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61 views

Kernel of the differential and weak topology

Suppose you have a differentiable map $\Phi : E \rightarrow F$ where $E$ and $F$ are Banach spaces, and a curve $t \mapsto u(t)$ of elements of $E$, with $u(0)=0$ and $\Phi(u(t))$ constant, such that ...
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627 views

Show that $c$ is closed in $l^{\infty}$

Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$ $$l^{\infty}=\{ (a_i)_{i \in \mathbb{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?
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Why isometric isomorphic between Banach spaces means we can identify them?

Is the "isometric" part really necessary? For what reason is that? Eg. we prove that there is an isometric isomorphism between $(L^p)'$ and $L^q$ ($(p,q)$ conjugate) and then we identify them ...
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1answer
268 views

Conditions for a kernel of a bounded operator to be complemented

I am well aware of the problem of complementing subspaces in Banach spaces as it was discussed here and here . Nevertheless, I wonder whether there are conditions for existence of a complement $M$ ...
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Proof that an embedding into $\ell^1$ is compact

Prove that any sequence $(x^{(n)})_{n\in\mathbb{N}}\subseteq\ell^1$ such that $\sum_{k=1}^\infty k\lvert x_k^{(n)}\lvert\leq1$ for all $n\in\mathbb{N}$ has a convergent subsequence. My thoughts ...
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1answer
171 views

Good source for Triebel-Lizorkin spaces?

I'm trying to look into different types of function spaces. At the moment, at least for function spaces involving integration, I only have $L^p$ and $W^{k,p}$. The next function spaces I thought I'd ...
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1answer
45 views

prove that $||d^2f(x)||\le M \Rightarrow ||df(x)||\le \sqrt{2Mf(x)}$

let E be a banach space , $f : E \to \mathbb R$ a function of $C^2$ / $f>0$ we suppose that $\exists M $ cte and : $||d^2f(x)||\le M $ prove that : $||df(x)||\le \sqrt{2Mf(x)}$
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1answer
93 views

Differentiation Operator a Contraction Mapping

Let $C^{\infty}[a,b]$ be the space of all infinitely differentiable functions on [a,b] with norm $$ || f || = \max _{[0,1]} | f(x) | , f \in C^{\infty}[a,b]$$ Is the differentiation operator ...
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224 views

Bounded operator on dense subspaces

Give an operator like this or show it doesn't exist: Operator $T: X\rightarrow Y$ is bijective. $X,Y$ are dense subspaces of a Banach space $Z$, and $X$ is proper subset of $Y$. Both $T$ and $T^{-1}$ ...
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Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
3
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1answer
317 views

$C_c^0(\Omega)$ is not Banach!?! Also density requires completeness?

Today I was very surprised to learn that $C_c^0(\Omega)$ is not a Banach space with the supremum norm. Why is that, when $C^\infty_c(\Omega)$ is? Also (from here, bottom of page 18), I learn that ...
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2answers
251 views

What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.

I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
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1answer
53 views

How to verify whether $(C_{00},\|\cdot\|_p)$ is complete

How to verify whether $C_{00}=\{(x_n):\text{All but finitely many terms are }0\}$ is complete with respect to $p$-norm given by $$\|(x_n)\|_p=\left(\sum_{n=1}^\infty|x_n|^p\right)^{1/p}$$ where $1\le ...
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139 views

Differential calculus on Banach space

I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly. Problem Given the ...
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151 views

Finite-dimensional Banach space

I've a problem with some exercise, namely: Show that if X is a finite-dimensional Banach space, then every linear functional f on X is continuous on X. Hint. Use Proposition: Every ...
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1answer
73 views

Need explanation of problem in Temam (convergence, weak derivatives)

Let $V \subset H \subset V$ be Hilbert triple. We have $u_m$ is infinite differentiable from $[0,T]$ to $V$. Suppose $u_m \to u$ in $L^2(0,T;V)$ and $u_m' \to u'$ in $L^2(0,T;V^*)$ Suppose that it ...
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84 views

On the existence of a non-negative function on a Banach space whose limit at every point is infinity.

Does there exist a Banach space $ X $ (possibly non-separable) and a mapping $ F: X \to X $ such that $$ \forall a \in X: \quad \lim_{\substack{x \in X \setminus \{ a \} \\ x \to a}} \| F(x) \|_{X} = ...
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1answer
48 views

Is this Nonlinear Autonomous Banach space valued ODE a flow?

I have the following analogue of Picard's theorem for Banach space valued ode's: Let $O$ be an open subset of a Banach space $B$ and let $F$ be a nonlinear operator satisfying the following criteria ...