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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34 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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1answer
33 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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1answer
39 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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1
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1answer
39 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 ...
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0answers
29 views
weak convergence in reflexive normed space
$X$ is a reflexive normed space. $\{x_n\}\subset X$ is a sequence bounded by $M$. Show that there exist a subsequence $\{x_{n_k}\}$ such that $x_n$ converges to $x_0$ weakly and $\|x_0\|\leq M$.
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1answer
16 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 ...
1
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1answer
34 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}
...
2
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0answers
54 views
Weakly closed subsets of $C(K)$
Given a compact Hausdorff space $K$, let us endow $C(K)$ with the Banach-space weak topology. Is there any handy description of weakly closed subsets of $C(K)$? Are subsets of $C(K)$ which are ...
2
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0answers
45 views
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 ...
2
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1answer
29 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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0answers
46 views
Does every Banach space admit a continous injection to a non-closed subspace of another Banach space?
Let $(V, ||\,||)$ be a Banach space. I want to produce a non-complete norm $||\,||'$ on it such that $||v||' \leq ||v||$ for all $v$ in $V$. Given a continuous injection $\varphi\colon V \to W$ with a ...
1
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0answers
50 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 ...
2
votes
3answers
83 views
Show that $c$ is closed in $l^{\infty}$
Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$
$$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ...
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0answers
45 views
What is $\lVert c \rVert_{X}$ where $c$ is constant?
A thought just occurred to me, in a Banach space $X$, what is
$$\lVert c \rVert_{X}=c\lVert \text{Id} \rVert_{X}$$
where $c$ is a constant? Is it even defined?
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votes
2answers
65 views
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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0answers
40 views
Example of nested sequence of non-empty closed convex sets in some banach space tha have empty intersection
I know tha if R - banach space, $B_1\supset B_2\supset\dots \supset B_n\supset\dots$ - sequence of nested closed balls in it, then it does have non-empty intersection. But is there an example of ...
3
votes
0answers
64 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$ ...
3
votes
2answers
61 views
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 ...
1
vote
1answer
26 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
38 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
30 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]$$
Is the differentiation operator ...
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0answers
63 views
Bounded operator on dense subspaces
Give an operator like this or show it doesn't exist: Operator $T: X\rightarrow Y$ is bijective. $X,Y$ are dense subspaces of a Banach space $Z$, and $X$ is proper subset of $Y$. Both $T$ and $T^{-1}$ ...
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1answer
93 views
$C_c^0(\Omega)$ is not Banach!?! Also density requires completeness?
Today I was very surprised to learn that $C_c^0(\Omega)$ is not a Banach space with the supremum norm. Why is that, when $C^\infty_c(\Omega)$ is?
Also (from here, bottom of page 18), I learn that
...
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1answer
34 views
What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.
I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
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1answer
33 views
How to verify whether $(C_{00},\|\cdot\|_p)$ is complete
How to verify whether $C_{00}=\{(x_n):\text{All but finitely many terms are }0\}$ is complete with respect to $p$-norm given by $$\|(x_n)\|_p=\left(\sum_{n=1}^\infty|x_n|^p\right)^{1/p}$$ where $1\le ...
4
votes
0answers
35 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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votes
2answers
51 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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votes
1answer
39 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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1answer
27 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
...
4
votes
1answer
44 views
A problem on bounded invertible linear operator in Banach space
Let $X$ be a Banach space. Let $T : X \to X$ be a invertible
linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all
$k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
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1answer
34 views
How to determine the spectrum on Banach Space [duplicate]
On Banach Space $C[0,1]$, T is a bounded linear operator and is defined by $Tf(x)=\int_0^xf(y)dy$, then how can I determine the spectrum of T?
I was hinted to first show $T^nf(X)=\frac1{(n-1)!} ...
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0answers
49 views
How we can change a strongly continuous semigroup to a contraction semigroup?
If $T(t), t>0$ is a strongly continuous, bounded semigroup on a Banach space. how we can transform it to contraction semigroup?
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0answers
25 views
canonical form of dyadic martingales
Let $(X_k)_{1\leq k \leq n}$ be a Walsh-Paley $L^p$-martingale (a dyadic martingale) with values in a Banach space $X$.
Why does there exist a dyadic martingale $(Y_k)_{1\leq k \leq n}$ with the ...
1
vote
1answer
29 views
Do I have a Banach space given this wacky norm?
I have a normed space (I'll denote it $C^2[a;b]$) which consists of continuous real functions whose first and second derivatives are also continuous in interval $[a;b]$.
$\forall x,y \in C^2[a;b]$ ...
8
votes
0answers
165 views
Infinite dimensional constant rank theorem
Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
1
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1answer
21 views
Is there any space with normal structure but not uniform normal structure?
It is clear that uniformly normal structure implies normal structure. But, is there any space that has normal structure but it doesnt have uniformly normal structure?? Would you mind to give me some ...
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0answers
39 views
Convex Hull of Precompact Subset is Precompact
I'm trying to prove that, if $K$ is a precompact (I've also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact.
I've come across a ...
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votes
2answers
44 views
Continuity in a Banach Space
This is a past exam question:
On the Banach space $\bigg(C\bigg( \bigg[-\dfrac{1}{2}, \dfrac{1}{2} \bigg], ||\cdot||_\infty\bigg) $, consider the operator given by
$$T(f)(x)= x+ ...
3
votes
3answers
55 views
Example of two norms on same space, non-equivalent, with one dominating the other
I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
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0answers
18 views
Lemma 3.2 from “Positive solutions for third order semipositone boundary value problems”
How de prove this lemma please :
Assume that: $w(t)$ is nondercreasing and $w(t)>0$ on $(q,1]$ , $\frac12<p<q<1$ hods .
Let $z\in C^2[0,1]\cap C^3(0,1)$ satisfy $z'''(t)\geq 0$ 0n ...
4
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1answer
98 views
Linear isometry between $c_0$ and $c$
The following question is an exercise and so I'm just looking for advices and not for answers if it's possible.
I have the following sets in $l^\infty$
$$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
4
votes
0answers
42 views
A space $X$ that contains a copy of $\ell_1$, does not contain a complemented copy of $\ell_1$, and whose dual is not weakly sequentially complete
I want to find an example of a Banach space $X$ which contains a copy of $\ell_1$, does not contain a complemented copy of $\ell_1$, and so that $X^*$ is not weakly sequentially complete.
1
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0answers
30 views
Definition of a countable direct sum of subspaces of a Banach space
Let $X$ be a separable Banach space and $K\subseteq X$ a subspace. Let $\{H_i\}_{i\in I}$ be a countable collection of subspaces of $X$. Is it correct that $K=\bigoplus H_i$ iff every element $k\in K$ ...
2
votes
1answer
57 views
Spaces of class $J_\alpha$
This question is about the spaces of class $J_\alpha$.
Given three Banach spaces $Z\subset Y\subset X$ (with continuous embeddings), and given $\alpha\in (0,1)$, we say that $Y$ is of class $J_\alpha$ ...
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0answers
45 views
$M+N$ is a closed subspaces of banach space iff $M^{\bot} +N^{\bot}$ is closed subspace of dual
Let $X$ be a Banach space and let $M,N$ be closed subspaces of $X$. I want to prove that $M+N$ is a closed subspace iff $M^{\bot}+N^{\bot}$ is a closed subspace of $X^{\ast}$ (i.e, dual of $X$).
Any ...
4
votes
2answers
60 views
Examples of Banach spaces
Which of the following are Banach spaces?
A. The set of all real-valued functions $f$, $g$ which are functions of an independent real variable $t$ and are defined and continuous on the closed ...
2
votes
1answer
26 views
Can a space have both a conditional and an unconditional basis?
Does there exist a Banach space $X$ which admits both a conditional and an unconditional Schauder Basis? If so, can one find an example in the collection of $\ell^p$ spaces?
My thoughts so far:
...
4
votes
1answer
51 views
Normed vector spaces and Banach spaces
Let $X$ be a Banach space with norm $||.||$ and let $S$ be a non-empty subset in $X$. Let $F_b(S,X)$ be the vector space of $F(S,X)$ of all functions $f:S \rightarrow X$ such that $\{||f(s)||:s \in ...
3
votes
2answers
62 views
Banach spaces and quotient space
Let $X$ be a normed vector space, $M$ a closed subspace of $X$ such that $M$ and $X/M$ are Banach spaces.
Any hint to prove that $X$ must be a Banach space?
3
votes
2answers
43 views
Show that the image of $T:l^{\infty}\to l^{\infty}$, $(x_n)_n \mapsto \Big(\frac{x_n}{n}\Big)_n$ is not closed in $l^{\infty}$.
Denote the set of all bounded sequences in $\mathbb{R}$ by $l^{\infty}$, endowed with the sup norm $\lVert \rVert _{\infty}$. Define a map $T:l^{\infty} \to l^{\infty}$ as follows:
$$(x_n)_n \mapsto ...
