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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Series Convergence in Banach Space

Let $(e_j)_1^\infty$ be an orthonormal set in $l^2$ Consider $$s_n =\sum_{j=1}^n t_je_j$$ Show that $s_n$ converges in $l^2 \iff t = (t_j)_{j=1}^\infty \in l^2$ Thoughts so far : If we consider ...
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34 views

Predual: Denseness

Problem Given a Banach space $E$. Regard a subspace: $$\iota:U\hookrightarrow E:u\mapsto u$$ Consider the projection: $$\pi:E'\twoheadrightarrow U':\psi\mapsto\psi\circ\iota$$ By Hahn-Banach find: ...
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2answers
135 views

completeness of cones in an ordered normed space

Let $(X,\|\cdot\|,\le)$ be a normed, ordered vector space over $\mathbb{R}$ and let $X^+=\lbrace x\in X:x\ge0\rbrace$ denote the (positive) cone in $X$. with a metric $d$ induced by the norm ...
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20 views

Stability for a infinite dimensional dynamical system

Suppose I have a infinite dimensional dynamical system as $\dot{x_n}=Ax$ where $A$ is an infinite-dimensional matrix and $\{x_n\}$ is a sequence of states. I was wondering if you can help me find ...
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1answer
26 views

Simple case of Young's inequality

I have a question concerning Young's inequality stated as follows: $||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$. Here you can find something on $\ell_q\big(\mathbb{Z}\big)$: Young's ...
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1answer
20 views

Codimension 1 closed subspace as a kernel of a functional

My non-linear analysis book says that if I have a linear operator $T:X\to Y$ with close range $R$ and $\operatorname{codim}(R)=1$ (and also $\dim(\ker(T))=1$) then there exists $\phi\in Y^{*}$ such ...
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36 views

Equivalence of norms in $C^1[0,1]$

i have the following problem/questions: I have to prove that $\lVert \cdot \rVert_1 \sim \lVert \cdot \rVert_{*} $ in $C^1[0,1]$; Where $\lVert \cdot \rVert_1$ is the usual $C^1[0,1]$ norm and ...
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23 views

Banach space and Hamel Basis cardinality

No infinite-dimensional normed linear space with a Hamel basis having cardinality strictly less than c can be complete. Can we prove it without using AC or Hahn-Banach Theorem?
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1answer
76 views

Hamel Basis in Infinite dimensional Banach Space without Baire Category Theorem

Prove that every Hamel basis in an infinite dimensional Banach Space is uncountable without using Baire Category Theory. We are assuming axiom that vector space dimension (if exist) is well-defined. ...
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1answer
47 views

A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$

A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$. How will we prove the converse implication. One sided implication for Hilbert Space is proved in ...
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33 views

Part of Lomonosov's Invariant Subspace Theorem

Let $X$ be a complex Banach space of infinite dimension, let $T\in\mathcal{B}(X)\backslash\{0\}$ be compact. Define $$\Gamma := \{S\in\mathcal{B}(X)\,|\,S\circ T=T\circ S\}$$and define, for each ...
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19 views

is every n-dimensional subspace of l2 isometrically isomorphic to l2n?

Let $E$ be an $n$-dimensional subspace of $\ell_2$. I seem to recall hearing that $E$ must be isometrically isomorphic to $\ell_2^n$, but I can't see why this would be the case, nor can I find a ...
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0answers
14 views

If $S$ and $T$ are closed vector subspaces then $S+T$ is closed [duplicate]

Let $V$ be a Banach normed space, $S,T \subset V$ be closed vector subspaces. Assume $\operatorname{dim}(T)<\infty$. Show that $S+T$ is closed. So I encountered this problem trying to use ...
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1answer
64 views

Brouwer fixed-point theorem infinite dimension [closed]

Brouwer fixed-point-theorem holds for compact convex set. Do you have example(s) where the theorem doesn't hold in infinite dimensional Banach spaces?
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1answer
24 views

Leray-Schauder fixed point theorem and remark

Leray-Schauder fixed point theorem from Gilbarg and Trudinger book is quoted below. I do not understand remark below this theorem. Could you explain? Theorem 11.2 from this text is Schauder fixed ...
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1answer
21 views

Question regarding convergence in $L^p$ spaces.

When solving an exercise regarding $\ell^p$ spaces, I came up with the following question. The exercise said, Let $1<p\leq \infty$ and let $p'=\frac{p}{p-1}$. Let $b\in \ell^{p'}$ and define ...
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1answer
34 views

Relation between two p-norms

While it's a well known that any two norms are equivalent for a finite dimensional normed linear space, I've been trying to derive the bounds for the case $X=\mathbb{R}^n$ and $l_p$-norms. Let $1 ...
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1answer
162 views

Proving $\ell_\infty$ is complete

I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on. For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
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0answers
55 views

Tensor product of bounded analytic functions

Let $H^\infty(\mathbb{D})$ denote the set of functions holomorphic and bounded on $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Conseqently, $H^\infty(\mathbb{D}^n)$ denotes the set of bounded ...
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22 views

Dense subsets in tensor products of Banach spaces [duplicate]

Assume $B_1$ and $B_2$ are Banach spaces of univariate functions. Moreover, assume that the sets $D_1 \subset B_1$ and $D_2 \subset B_2$ are dense with respect to the respective norms ...
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1answer
52 views

Dense subsets in tensor products of Banach spaces

Assume $B_1$ and $B_2$ are Banach spaces of univariate functions. Moreover, assume that the sets $D_1 \subset B_1$ and $D_2 \subset B_2$ are dense with respect to the respective norms ...
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1answer
31 views

Does any continuous map between two Banach spaces over a nonarchimedian field is a closed map?

Does any continuous map between two Banach spaces is a closed map? It seems to me that it is true. Let $f:V \rightarrow W$ be a map, where $V$, $W$ are (infinite dimensional) Banach space over a a ...
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2answers
47 views

Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint

Question:Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint. Suppose that the spectrum $\sigma(a)$ is an infinite set. Show that $A$ is infinite-dimensional. How can i prove it? I guess: Let $A$ be ...
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1answer
81 views

A copy of $l_\infty$ in a infinite dimensional Banach space

Let $E$ an infinite dimensional Banach space. Using the Hahn-Banach extension theorem, prove that there is a sequence $(y_n)\subset E$ and a decreasing sequence of closed subspaces ...
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28 views

Spectrum of integration operator on $C[0,1]$.

I'm trying to find the spectrum of the operator $T: C[0,1] \to C[0,1]$ given by: $$T(f)(t) = f(0) + \int_0 ^{t} f(s) ds$$ I can show that $0$ is contained in the approximate point spectrum with ...
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1answer
20 views

Double annihilator of subspace of dual space

If $X$ is a Banach space then it's quite straightforward to show that for $A$ a subspace we have $\bar{A} = {(A^{\circ})}_{\circ}$ and so if $A$ is finite dimensional then $A = {(A^{\circ})}_{\circ}$. ...
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14 views

C*−algebras and operator theory, murphy [duplicate]

Do you know how to solve this exercise? (Murphy, $C*$−algebras and operator theory, $2^{nd}$ chapter, 1st exercise) Let $A$ be a Banach algebra such that for all a\in A the implication $Aa=0$ or ...
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14 views

Prove map is inflating

Let $T:X\to Y$ be a continuous linear open map between two Banach spaces. Prove that $\exists K\in\mathbb{R}$ such that for each $y\in T(X)$ we have $$T^{-1}(\{y\})\cap B_{K||y||}(0)\neq\varnothing$$ ...
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19 views

Second Order Mean Value Inequality In Banach Space

I have some confusions about proving the following theorem from Luenberger's Vector Space Optimization book, Proposition 2 p.176: $\textbf{Claim:}$ Let $X$ be a vector space and Y be a normed space. ...
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1answer
29 views

A problem which reverses the definition of a bounded operator

I've encontered a problem that appears simple, almost like it's a definition of a bounded operator, but with a reversed inequality sign... and I can't seem to find my way to a solution. Any ...
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3answers
946 views

Applications of the Hahn-Banach Theorems

Question: What are some interesting or useful applications of the Hahn-Banach theorem(s)? Motivation: Most of the time, I dislike most of Analysis. During a final examination, a question sparked my ...
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example of the arens multiplication; I want to unterstand the construction

I want to understand the double dual as a $C^*$-algebra of a given $C^*-$algebra $A$ but my first problem is to understand the arens multiplication on the double dual $A^{**}$ (considered as Banach ...
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4answers
57 views

Show that if $\sum x_n$ converges then $x_n \to 0$

Let $(V,\|\|)$be a normed space. Let $(x_n) \subset V^{\Bbb{N}}$. We say that $\sum x_n$ converges if, $\lim_{n\to \infty} \sum_{i=1}^{n}x_i$ exists. Show that if $\sum x_n$ converges then ...
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Is a Banach space also a metric space?

Since a Banach space is a complete normed vector space and a norm always induces a metric, a Banach space must be a metric space, right? If so, why is a Banach space defined as a complete normed ...
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41 views

Calculating the generator of a weighted transition function

Let $(P_t)_t$ be the transition function of a Feller-Dynkin process $X$. The usual Banach space of functions that the semigroup $(P_t)_t$ is working on is $C_0(E)$, i.e. continuous functions that ...
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Showing projection is continuous if and only if kernel is closed

I have a linear map $P$ on a Banach space, $X$, with $P^2 = P$ and I'm trying to show that $P$ is continuous if and only if $\ker(P)$ and $\ker(I-P)$ are closed. One direction is straight forward but ...
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1answer
9 views

Determining if linear operator on space of polynomials is bounded

I have $p$ a polynomial given by $p(x) = a_0 + a_1 x + a_2 x^2 ... a_n x^n$ and a linear operator $T$ defined by $T(p)(x) = a_0 + a_1 x^2 + a_2 x^4 + ... + x^{2n}$. The norm on the space is given by ...
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1answer
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A question in Banach space. [closed]

Let $C$ be the Banach space of all complex continuous functions on $[0, 1]$, with the supremum norm. Let $B$ be the closed unit ball of $C$. Why there exist continuous linear functionals $\Gamma$ on ...
2
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2answers
41 views

Prove $c_0$ is a banach space.

The subspace of null sequences $c_0$ consists of all sequences whose limit is zero. Prove that $c_0$ is a closed subspace of $C$ (The space of convergent sequences), and so again a Banach space. ...
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1answer
18 views

Differentiability in normed spaces

I really need a help with the following exercise: Suppose $\mathbb{E}$ and $\mathbb{F}$ are normed spaces, $A \subseteq \mathbb{E}$ is an open set, $f: A \to \mathbb{F}$ is differentiable on $A$, and ...
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A Counter Example about Closed Graph Theorem

This is an example I read around closed graph theorem. Let $Y=C[0, 1]$ and $X$ be its subset $C^\infty[0, 1]$. Equip both with uniform norm. Define $D: X \to Y$ by $f \mapsto f'$. Suppose $(f_n, f_n') ...
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Finding isometries of a Banach Spaces.

Given a Hilbert Space $(H,\langle,\rangle)$, $x,y\in H$ and $D\subset H$ a subspace of $H$ (I mean, the operators $+$, $\cdot$ and $\langle,\rangle$ in D are the restrictions of the respective ones in ...
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1answer
32 views

About closed graph theorem

I want to show that in the closed graph theorem, the completeness of $Y$ is essential. (a.e I want to find two norm space $X,Y$ which $Y$ isn't complete and linear function $T:X\to Y$ such that $T$ is ...
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Characterization of Bochner dual

I want to prove following theorem Let X be separable and reflexive Banach space, $1<p<\infty$ than $$ L^p((0,1),X)^* = L^q((0,1),X^*) $$ where $\frac1{p}+\frac1{q} = 1$, with ...
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1answer
116 views

Proving identity of a $C_0$ semigroup

We have a strongly continuous semigroup $\{T(t)\}_{t\ge 0}$ on a Banach space X with a generator $ A:D(A)\subset X\to X$. I need to show that $\displaystyle T(t)x = \sum\limits_{n=0}^{\infty} ...
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27 views

$L^2-$summand vectors and Paralellogram Law in real Banach spaces

Let $X$ be any real Banach space and $p\in X$, then the Parallelogram Law holds, trivially, for every couple $u,v\in span\{p\}$. We say that $x\in X$ is an $L^2$summand vector of $X$ if ...
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3answers
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prove that $C_0(X)$ is banach space .

For prove that $C_0(X)$ is banach space X is vector space with norm $||f||_{\infty}$ . I'm trying to prove that $C_0(X)$ is close subset of $C(X)$ therefor i suppose $f \in \overline{C_0(X)}$ so there ...
4
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1answer
52 views

How to prove that $(C[a,b], \|\cdot\|_\infty)$ is not a reflexive Banach Space [duplicate]

The tag line basically says it all...this is a question in Luenberger's Optimization book (5.14.4 on p.138). Clearly I don't expect someone to deliver a full proof if it's tedious, but a sketch or ...
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Example of non-reflexive Banach space

How does one prove that $C^0([0;1],\mathbb{R})$ equipped with the sup norm is not reflexive? I don't understand how to show that the $J$ mapping is not surjective.