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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2
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1answer
23 views

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
-2
votes
0answers
23 views

normed spaces and their completion [on hold]

Let $V$ a normed space and $(W,i)$ their completion. Let $U$ a Banach space and $T:V\rightarrow U$ a continuous linear operator. $(a)$ Prove that exists an unique continuous linear operator $ ...
0
votes
1answer
12 views

Codimension of the image of the polynomials subspace is infinite

Consider the interval $I=[0,1]$ and the Banach space $E$ of real continuous functions defined on $I$ ($E=\mathcal C_{\mathbb R}(I))$. $P \subset E$ is the subspace of polynomial functions (restricted ...
-2
votes
1answer
42 views

L(V,W) is Banach, then W is Banach [duplicate]

Let $V,W$ normed vector spaces, $V$ not empty and with a finite dimension. Prove that $L(V,W)$ is Banach, then $W$ is also Banach.
1
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1answer
19 views

How is completeness used in the Eberlein–Šmulian theorem?

So for weakly closed subsets of Banach spaces compactness and sequential compactness coincide, but upon studying the proof I can't put my finger on what exactly would go wrong if we dropped the ...
1
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1answer
28 views

Show that Quotient of a C*-algebra is a C*-algebra in its own right

Let $A$ be a commutative C*-algebra over $\mathbb{C}$ and let $J$ be an ideal of $A$ such that $J\in Closed(A)$ and that contains $x^*$ if it contains $x$, for all $x\in A$. I know that $A/J$ is also ...
3
votes
1answer
50 views

How to show that the space of polynomials is not complete

Denote by $P[0,1]$ the set of all polynomials $p\colon [0,1]\to\mathbb{R}$; this is a vector space. Endow $P[0,1]$ with the norm $$\| p\|=\sup_{t\in [0,1]}{| p(t)|}.$$ I want to show that this ...
0
votes
0answers
18 views

Why's Daugavet equation important?

I've been recently studing Daugavet equation in $L^1[0,1]$ and $C[0,1]$. I understand most of the results I've found but I can't figure out why is it important to find operators that hold Daugavets ...
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0answers
28 views

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 ...
0
votes
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 ...
2
votes
0answers
25 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 ...
1
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1answer
21 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 ...
0
votes
0answers
28 views

Banach space and Hamel Basis cardinality

No infinite-dimensional normed linear space with a Hamel basis having cardinality strictly less than $\mathfrak c$ can be complete. Can we prove it without using AC or the Hahn-Banach Theorem?
3
votes
0answers
37 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 ...
2
votes
0answers
39 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 ...
2
votes
1answer
49 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 ...
0
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0answers
20 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 ...
0
votes
1answer
77 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. ...
0
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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 ...
3
votes
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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votes
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 ...
2
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0answers
56 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 ...
2
votes
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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0answers
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 ...
0
votes
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 ...
3
votes
2answers
48 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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0answers
29 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 ...
0
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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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0answers
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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0answers
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 $$T^{-1}(\{y\})\cap B_{K||y||}(0)\neq\varnothing$$ ...
0
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0answers
21 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 ...
2
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0answers
16 views

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 ...
2
votes
4answers
60 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 ...
-1
votes
1answer
28 views

${x_n} \to x$ weakly, why does $T{x_n} \to Tx$ weakly? [closed]

If $T \in B(X,Y)$ and ${x_n} \to x$ weakly, why does $T{x_n} \to Tx$ weakly?
0
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3answers
61 views

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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0answers
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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0answers
30 views

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 ...
0
votes
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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votes
1answer
39 views

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
votes
2answers
42 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
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: ...
1
vote
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 ...
2
votes
1answer
33 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 ...
2
votes
0answers
31 views

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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0answers
24 views

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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0answers
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 ...
2
votes
3answers
43 views

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
votes
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 ...
0
votes
3answers
38 views

Is this space complete or is it incomplete?

Show (if possible) that the space of all complex sequences $x=(x_n)$ with only a finite number of terms nonzero (the number of nonzero terms may be different for different members of the space) is ...