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.

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0answers
27 views

$ \frac{1}{k} \sum_{i = 1}^{k} {g_{i}} \to g_{\infty}$ weakly a.e, imply that $\frac{1}{k} \sum_{i = 1}^{k} {g_{i}} \to g_{\infty} $ in measure?

Let $(E,\mathcal{A},\mu)$ be a finite measure space and $(X,\|.\|)$ be a reflexive Banach space. The en set of all Bochner-integrable function from $E$ to $X$ is denoted by $\mathcal{L}_{X}^{1}$. $(...
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1answer
87 views

why $x_m$ converges weakly to $x_\infty$?

Let $(X,\|.\|)$ be reflexive Banach space and $Y$ be a closed separable subspace of $X$ $\big((Y ,\|.\|)$is clearly a separable reflexive Banach space$\big)$, then the dual space $Y^*$ of $Y$ is ...
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2answers
17 views

Showing that $G$ is sequentially closed in $X^*$ with the $\sigma(X^*,X)$ topology.

Let be $X$ Banach space. Let $Y$ be a linear subspace of $X^*$ such that $Y$ is dense for $\sigma(X^*,X)$. Let $\sigma(Y)$ be the sigma algebra on $X$ generated by sets of the form $$\{x\in X:(\langle ...
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3answers
170 views

Can this complete metric space be a Banach space?

Let $(S,d)$ be the space of all sequences in $\mathbb{R}$ with the metric $$d(\mathbf{x},\mathbf{y})=\sum_{i=1}^{\infty}\dfrac{1}{2^i}\dfrac{|\xi_i-\eta_i|}{1+|\xi_i-\eta_i|}$$ where $\mathbf{x}=(\...
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1answer
12 views

Proving a linear form to be continuous

In the Hilbert space $L^2(R)$ I have seen that the following form is linear, however, I need to check if it is continuous and find the associated vector using the Riesz-Fréche Theorem. I have tried to ...
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1answer
50 views

Find it adjoint $T^{*}$ [closed]

Show that the operator $$T: \ell_{2} \rightarrow \mathbf{C},Tx := \sum_{n=1}^{n=\infty} \frac{1}{n}x_{n}$$ is bounded.Then find it (Banach) adjoint $T^{*}$.
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16 views

Biconjugate formula in non-separated locally convex spaces

I have a locally convex space $X$ with topological dual $X^*$ and coupling $\langle x,x^* \rangle:=x^*(x),\ x\in X,\ x^*\in X^*$. For $f:X\to\overline{\mathbb{R}}$ one defines its convex conjugate ...
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1answer
18 views

Closed ball is weakly closed

The problem is in a Banach space, if $||x_n||\leq 1$ and $x_n\to x$ weakly, then $||x||\leq 1$ This question has an answer here: math.stackexchange.com/questions/714049/closed-unit-ball-in-a-banach-...
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0answers
15 views

Proof of several facts on bounded operators $A : X\to Y, B : Y \to X$ such that $AB$ is Fredholm

Let $A : X\to Y$ and $B : Y \to X$ be bounded operators between Banach spaces $X,Y$. Assume that $AB : Y\to Y$ is Fredholm. I would like to prove that $A(X)$ and $B(Y)$ are closed. Furthermore, ...
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1answer
36 views

$X$ Banach, $T(X\to X)$ with compact resolvent implies $T+B$ is Fredholm with index $0$ provided $B$ is bounded

I am trying to solve this problem from Gohberg's book: Let $X$ be a Banach space and suppose that $T : \mathcal{D}(T) \subset X \to X$ has compact resolvent. Then, if $B$ is bounded on $X$, the ...
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10 views

Using implicit function theorem on Banach spaces to solve equations

I have seen it mentioned in the literature that one can often deal with a (quasilinear) non-linear PDE or a system of non-linear PDEs by perturbing to a linearised system and then finding solutions ...
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2answers
32 views

Power of a bounded linear map nullify each point implies the power is eventually zero

I am working on a problem from my Qual "Let $T:V\to V$ be a bounded linear map where $V$ is a Banach space. Assume for each $v\in V$, there exists $n$ s.t. $T^n(v)=0$. Prove that $T^n=0$ for some $n$....
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1answer
83 views

Every separable Banach space is isometrically isomorphic to a quotient of $\ell^1$

I am currently trying to prove the statement above. So let $X$ be a Banach space and choose a dense sequence $(x_n)_n$ in the closed unit ball of $X$. Then it is easy to see that $$T: \ell^1 \to X, \...
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1answer
24 views

Practical Applications of the Fréchet Derivative

I understand the definition of the Fréchet derivative. However, outside of functions on $\mathbb R^n$, I've never encountered an application where it was particularly useful. Can anyone share ...
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1answer
35 views

Alternative proof that of the Uniform Boundedness Principle when $Y$ is a Banach space

I just read an alternative proof of the Uniform Boundness Principle when $Y$ is banach space that goes like this; Suppose we have $X$ and $Y$ Banach spaces and $F \subset L(X,Y)$ such that for every $...
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0answers
43 views

Application of Banach fixed theorem for differential equations

Let $f \in C([0,1] \times \mathbb{R})$ and there exists $0 \leq \gamma < 8$ such that for any $0 \leq x \leq 1, u,v \in \mathbb{R}$, $$|f(x,u)-f(x,v)| \leq \gamma|u-v|.$$ Let $\alpha,\beta \...
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1answer
36 views

Space of continuous functions form a Banach space?

Let $X$ be the collection of all continuous real-valued functions defined by \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\vert } \end{equation*} ...
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1answer
30 views

Pointwise Convergence in Banach Space Implies Convergence in Operator Norm

Assume that $(a_n : V \rightarrow W, n \geq 0)$ is a sequence of continuous linear maps with $V$ is Banach space, $W$ a normed space such that $(a_n(v))_{n \geq 0}$ is convergent for any $v \leq V$. ...
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11 views

Completion of a norm space, proof understanding.

I just came across a proof that shows that each normed spaces can be completed to a Banach space. I would love to get some things straightned out though. This is the proof: https://prnt.sc/shl2f4 ...
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1answer
62 views

Weak convergence in reflexive space is equivalent to singleton intersection of the convex hulls

Question Let $X$ be a Reflexive space. Suppose $(x_n)_n \in X$ is a bounded sequence and define $K_n=\overline{conv\{x_m,m\geq n\}}$. Then $x_n\overset{w}{\to}x_0$ if and only if $\cap_{n\in\mathbb{...
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1answer
34 views

Reference on separability of $c_0(X)$, where $X$ is a separable Banch space

Let $(X,\|\cdot\|)$ be a separable Banach space. By $c_0(X)$ I mean the space $\{(x_n)_n\subset X:\, \|x_n\|\to0\}$. I think it os well known that $c_0(X)$ is a separable Banach space endowed with ...
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1answer
35 views

Banach Space Inequality of functionals equivalence

Let $X$ be a Banach space with $f_1,..,f_n\in X^*$ and $c_1,...,c_n\in\mathbb{R}$ then the following are equivalent: 1. $\exists x_0\in X: f_i(x_0)=c_i,\forall i\in\{1,..,n\}$ 2. $\exists M\geq 0: |\...
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0answers
39 views

Basic sequence weak convergence implies convergence in norm

A sequence $(x_n)_n$ of a Banach Space $X$ is called basic if it is a basis of $\overline{span\{x_n\}}$. Prove that if $x_n\overset{w}{\to}0$ then $\|x_n\|\to 0$. I was trying to define a $T\in X^*$ ...
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0answers
15 views

Coervive Map on a Banach Space

If $B$ is a finite-dimensional Banach space then norm-coercivity and coercivity coincide since the weak and strong topologies coincide. However, if $B$ is infinite-dimensional, say $B=L^p$ for $p\in [...
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1answer
64 views

Proving $\| g^{-1}-f^{-1}\|\leq 2\|{f^{-1}}\|^2\|g-f\|$ [duplicate]

Let $f,g\in L(X,Y)$ where $X,Y$ are Banach spaces and let $f^{-1},g^{-1}\in L(Y,X)$. How to prove that $$\| g^{-1}-f^{-1}\|\leq 2\|{f^{-1}}\|^2\|g-f\|$$ if $\|g-f\|\leq\frac{1}{2\| f^{-1}\| }$?
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2answers
34 views

The Relationship between Reflexive Space, Separable Space and Compactness

I'm currently studing functional analysis. It bothers me a lot about the relationship between reflexive space, separable space, and compactness of unit ball (in different space, in different topology)....
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2answers
45 views

Showing incompleteness of metric space by specifying non-convergent Cauchy sequence

I have been looking at the space of continuous functions over a compact interval $C([0,2])$ equiped with the the integral norm of absolut values $\| \cdot \|_1$. I read a counterexample that showed ...
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1answer
34 views

Canonical projection on Quotient space maps open ball to open ball

I’m struggling to understand the last part of this proof where it says $\pi(x)=\pi(x-z)$ proves the claim. To prove the claim I suppose this must imply that $||x|| \leqslant ||x-z||$ so that $||x||<...
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12 views

Construct a vectors and dual vectors with given property

Let $X$ be an infinite-dimensional normed space (can assume Banach if needed). Given an integer $n\in\mathbb{N}$ and a mapping $f:\{1,\dots,n\}^2\to\{\pm 1\}$, I want to construct $\{x_1,\dots,x_n\}\...
3
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1answer
69 views

Unbounded linear operator between normed spaces

I am in the middle of a proof and this is one step I don’t understand Let $T:E\rightarrow F$ be a linear operator between normed spaces $E$ and $F$ If $T$ is unbounded then there exists a sequence $...
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0answers
11 views

Algebraically complemented subspace

I'm studying algebraically complemented subspace in Banach space and as the definition of it I have to suppose that also its complemented has to be close, but I don't understand why, I know it is a ...
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0answers
56 views

Is the space of all bounded linear operators form a Banach space (under the given norm)?

Consider the class of all bounded linear operators on $X$ such that for each $f$ we have \begin{equation*} m(f)=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\vert }. \end{...
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0answers
27 views

Compact embedding from Lipschitz set of functions into continuous set

We note $E=\Biggl\{f:[0,1]\to\mathbb{R},~\sup\limits_{x\neq x'}\frac{\vert f(x)-f(x')\vert}{\vert x-x'\vert^\frac{1}{2}}<\infty\Biggr\}$ I have showed that $(E,\Vert\cdot\Vert_E)$ is a Banach ...
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0answers
32 views

Every strictly singular operator on $l_2$ is compact

An operator $T:X\to X$, where $X$ is a Banach space, is strictly singular if no restriction to a closed, infinite dimensional subspace is an isomorphism. It is well known they form a closed ideal in $...
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1answer
39 views

A metric for weak* convergence on bounded sets

Let $X$ be a separable Banach space and let $(x_n)_{n \in \mathbb{N}}$ be a fixed sequence such that $x_n \neq 0$ for all $n \in \mathbb{N}$ and $\lbrace x_n: n\in \mathbb{N}\rbrace$ is dense in $X$. ...
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1answer
35 views

Logarithm of the norm of a Banach space-valued holomorphic function is subharmonic?

Given a holomorphic function $f$ in some open subset $G\subset\mathbb{C}$, it is well-known that the real-valued function $z\in G\mapsto\log|f(z)|\in\mathbb{R}$ is a \emph{subharmonic} function, i.e., ...
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1answer
24 views

About bounded operator in normed space

Suppose that $T$ is a linear operator from a normed space $X$ into a normed space $Y$ such that $\sum_{n=1}^{\infty} T (x_{n})$ is a convergent series in $Y$ whenever $\sum_{n=1}^{\infty} x_{n}$ is ...
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0answers
32 views

Space of Lipschitz functions (with some conditions) form a Banach space

Let $X=[0,1]$ and \begin{equation*} L = \lbrace f:X\rightarrow \mathbb{R}:f \ \text{is Lipschitz}, f(0)=0 \ \text{ and }\ f(1)=1\rbrace \end{equation*} with norm \begin{equation*} \|f\|=\sup_{x\neq ...
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0answers
55 views

What is the geometrical meaning of the space?

Consider the Banach space of all lipschitz functions on $X$ such that for each $f$ in the space, \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\...
3
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3answers
70 views

How to prove that whether it is a Banach space or not?

We consider the Banach space of all continuous functions on $X$ such that for each $f$ in the space, \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\...
0
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1answer
44 views

Convergent subsequence of a given sequence

Sorry, my initial problem statement was weird. I am correcting the problem statement. The following is the precise statement of the problem: Let $D$ denote the set of functions (continuous) in $C([0,...
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0answers
11 views

Isomorphism between $\mathcal{l}_p(E) '$ and $\mathcal{l}_q(E')$

Let $1 \leq p < \infty $, $1/p+1/q=1$ and $E$ a Banach space. Define $\mathcal{l}_p(E) = \{ (x_n)_{n=1}^{\infty}\,: \, x_n \in E\,\,\, \forall\,\, n \in \mathbb{N}\,\,\, \mbox{e}\,\,\, || (x_n)_{n=...
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0answers
17 views

Show that: for all $x^*\in D$: $ \lim_m\frac{1}{m}\sum_{n=1}^{m}{\sup_{x\in A_n^k}{\langle x^*, x\rangle}}=\beta_{x^*}^k $

Let $X$ be a separable Banach space. $X^*$ will denote the dual of $X$ and $\langle,\rangle$ the usual duality. Put: $$ \mathcal{P}_{c}=\{C\subset X:C\text{ is nonempty closed convex subset of }X\} $$ ...
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1answer
28 views

Open cover of $A\subseteq\mathbb{R}$ such that it does not have a finite subcover.

If $A\subseteq\mathbb{R}$ is not closed. How can I build an open cover of $A$ such that it does not have a finite subcover?
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0answers
31 views

Spectrum of restriction to invariant subspace

Let $H$ be a separable Hilbert space and let $\mathcal{B}(H)$ denote the algebra of linear bounded operators on $H$. Let $T \in \mathcal{B}(H)$ and let $M$ be a non-trivial closed invariant subspace ...
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1answer
50 views

Textbook recommendations for weak topology

This is not a question regarding a specific mathematics problem, rather I am looking for some good texts that go into detail on the weak topology. My exposure to weak topologies is via Banach spaces ...
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1answer
29 views

Distance to the kernel of a functional in Hilbert space

I need to prove that in Hilbert space $H$ the distance between point $z$ and the kernel $L$ of a linear functional $f$ is $d(z,L)=\frac{|f(z)|}{\|f\|}$. I know a rather sophisticated proof for Banach ...
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0answers
22 views

Extreme point of $B_A = \{a\in A: \|a\| \leq 1\} $ in a Banach algebra $A$

I've send this question a little while ago now I edit it Let $A$ be a Banach algebra with identity $e_A$ prove that $e_A$ is an extreme point of $B_A = \{a\in A: \|a\| \leq 1\} $ Recall ...
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0answers
21 views

Algebraic dimension of the dual of $\ell^{\infty}$

It is shown that the algebraic dimension of an infinite dimensional Banach space is equal to it's cardinality. Using this fact, can we say that the algebraic dimension of $(\ell^{\infty}(X))^*$is ...
1
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1answer
80 views

Prove $e_A$ is an Extreme point of $B_A = \{a\in A: \|a\| \leq 1\} $ in a Banach algebra $A$

Let $A$ be a Banach algebra with identity $e_A$ prove that $e_A$ is an extreme point of $B_A = \{a\in A: \|a\| \leq 1\} $ Recall that $x$ is an extreme point of $B_A$ if $x = \frac{1}{2}(x_1+...

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