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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C*-algebras: States?

I'd like to better understand states on C*-algebras. What properties should I investigate and in which order? (Positive functionals, extremal states, Schwarz's inequality, Kadison's inequality, what ...
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16 views

Density and Fredholmness

Let $X$ be a Banach Space and $Y$ a dense subset of $X$. An operator $T:X \to X$ is said to be Fredholm if it has closed range, $\dim \ker(T)<\infty$ and $\dim coker(T) < \infty$. Here is my ...
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24 views

The dual of the space $L^\infty$. [duplicate]

As we know the dual of $L^p$s are $L^q$s where $\frac{1}{p} + \frac{1}{q} =1$, and dual of $L^1$ is $L^\infty$. What is dual of the space $L^\infty (E)$ where E is a measurable subset of $\mathbb{R} ^ ...
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15 views

Geometric intuition behind of uniformly rotund in every direction

The norm of a Banach space $X$ is said to be uniformly rotund in every direction if $\lim_{n\to\infty} ||x_n-y_n||=0$ whenever $x_n, y_n \in S_X$ are such that $\lim_{n\to\infty} ||x_n+y_n||=2$ and ...
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42 views

Fundamental theorem of calculus with Gâteaux differentials and Riemann integrals

Let $f:[a,b]\to E$ where $E$ is a Banach space and let $Df(x,h)$ be its Gâteaux differential in $x$ with direction $h$. If $\mathbb{R}\to E$, $h\mapsto Df(x,h)$ is linear and continuous, then we write ...
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1answer
57 views

Having trouble understanding a proof after it applies the Hanh Banach theorem.

I have been reading a proof on the convergence of Newton's method that has been fairly easy to follow except for a single step that has totally mystified me because it suddenly depends on a lot more ...
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1answer
22 views

Homeomorphism from $K$ to $\Phi_{C(K)}$

Let $K$ be a compact Hausdorff and $C(K)$ be a Banach algebra of continuous function on $K$ such that $\textbf{1} \in C(K)$ and such that $C(K)$ separates the point of $K$. I am trying to show that ...
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24 views

Spectrum of a bounded operator on a (not necessarily Banach) normed vector space

It's well known that on a Banach space, the spectrum of each bounded operator is compact in $\mathbb C$. What about a general normed vector space? Is there a counterexample if we don't assume ...
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50 views

Closed subspace. A Hahn–Banach theorem consequence

I am trying to prove: If M is a subspace of a normed space $X$, that $\overline{M}=\bigcap\{\ker(\phi):\phi|_{M} = 0 \}$ It is really easy to see that $\overline{M} \subset ...
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37 views

Smallest constant of Lipschitz retraction from bounded to continuous functions

Let $B$ be the space of all bounded functions $f:[0,1]\to\mathbb R$ equipped with the supremum norm*. It contains $C$, the space of continuous functions on $[0,1]$, as a subspace. An $L$-Lipschitz ...
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40 views

Newton method for maps between Banach spaces

I am trying to understand the following theorem, which can be found in Kolmogorov and Fomin's (p. 509 here): Let map $F$ [$:X\to Y$ where $X,Y$ are Banach spaces] be strongly differentiable in a ...
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135 views

defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
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20 views

Reference about $p$-homogeneous functions

I'm looking for a book about $p$-homogeneous functions. I am particularly interested in the associated (nonlinear) eigenvalue problems. However, a reference containing most of the known properties of ...
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1answer
28 views

Dynamics: Schwinger-Dyson-Expansion

Given a C*-algebra $\mathcal{A}$ Consider a free generator $\delta_0:\mathcal{D}_0\to\mathcal{A}$ with $\overline{\mathcal{D}_0}=\mathcal{A}$. Introduce a perturbation ...
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1answer
21 views

Lower bound for the norm of the resolvent

I need to prove next statement (I want to do it for general case) $\|R_A(z)\| = \lVert \frac{1}{A-zI} \rVert \ge \text{dist}(z,\sigma(A))^{-1}$ I think it could be like this let $a\in \sigma(A) z ...
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48 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
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44 views

Subalgebras of $C_b(X)$ whose elements do not vanish simultaneously at any point

Let $X$ be a completely regular space. How can I find all Banach subalgebras of $C_b(X)$ (all complex-valued bounded continuous functions on $X$) with the property that for every $x\in X$ there ...
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22 views

Banach Algebra spectral theory [on hold]

Let $(\Omega, \mu)$ be a measure space. Show that the linear span of the idempotents is dense in $L_\infty(\Omega, \mu)$.
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1answer
34 views

Extreme point of unit balls, over $\mathbb C$

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $\mathcal{C}[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...
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35 views

Extreme point of unit balls, the complex case [duplicate]

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $C[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...
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1answer
27 views

How to prove that operators are isometry on $\ell^p$/$c$?

Lately I've been studying Banach spaces and isometries, and encountered many explicit isometrys involving $c$, $c_0$, $c^*$, $c_o^*$, $\ell^1$, $\ell^\infty$, etc... ($c \subset \ell^\infty$ is the ...
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Strong derivative of a compound map

I find the strong Fréchet derivative of $\Phi(h,\psi(h))$, where $\Phi:T_0\times T_\xi\to Y$ with $T_0, T_\xi, Y$ Banach spaces and $\psi:T_0\to T_\xi$ is strongly differentiable in $0$, evaluated in ...
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31 views

Is a countable union of complete subspaces complete?

I would like to ask the following, which I wanted to use a part of my proof but couldn't determine if it's right: Assume $X$ is a normed space, and $(X_n)_{n\in \mathbb N}$ complete subspaces. Must ...
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+100

Eigenvalues of Left Shift + Right Shift in $l_2([0,\infty))$

This question appeared on an old final exam and I am having difficulty completing it for practice. Let $S_r$ and $S_l$ be defined on the hilbert space $l_2[0,\infty)\to l_2[0,\infty)$ as the ...
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60 views

Continuity of implicit function

I find a proof of the following theorem in A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа (pp. 492-493 here): Let $X,Y,Z$ be Banach spaces, $U$ a neighbourhood ...
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1answer
36 views

The exponential of the identity operator in a Banach space

Let $X$ be a Banach space and $I \in L(X)$ be the identity operator. Determine the action of the operator $e^I$ on $X$. Pretty stuck here, not sure exactly what it means by determine the action. ...
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1answer
40 views

Is the Product of Banach Spaces a Banach Space?

Let $X$ and $Y$ be two Banach spaces (not necessarily possessing the same norm). The product space $X×Y=Z$ is given the max norm, i.e. $\max(\Vert x\Vert, \Vert y\Vert)$, where $x$ is given the norm ...
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24 views

Let $X$ and $Y$ be Banach, and $T : X \to Y$ a bounded linear map. When is $T$ closed?

I know that $||T(x)|| \geq C ||x||$ for some $C$ is equivalent. I am looking for less analytic conditions, maybe something to do with the topological structure of $X$. Does anyone know some? By $T$ ...
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38 views

CAR- & Weyl-Algebra: Uniqueness?

Given a Hilbert space: $\mathcal{h}$ Consider representations of the CAR-algebra: $\mathcal{A}_\text{CAR}^{(\prime)}(\mathcal{h})$ In Bratelli & Robinson it is stated the uniqueness: ...
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Why should the open mapping theorem be expected?

Soft question alert. I want to know why to expect the open mapping theorem to be true. My thoughts: I know that one nice consequence of the OMT could be thought of as the universal property of ...
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1answer
39 views

Are all Banach spaces also Hilbert spaces?

We have the well-known "polarization identity" $$(x,y)=\frac{1}{4}\left(\|x+y\|^2-\|x-y\|^2+i\|x+iy\|^2-i\|x-iy\|^2\right)\tag{1}$$ that works in any Hilbert space. Hence, is every Banach space also a ...
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46 views

Riemann integrals of abstract functions into Banach spaces

If we define the (Riemann) integral of an abstract function, i.e. a function $f:[a,b]\to Y$ where $Y$ is a Banach space, as$$\int_a^b F(t)dt:=\lim_{\max(t_{k+1}-t_k)\to ...
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13 views

Prove that a complex-valued homomorphism on a Banach algebra which is not identically 0, is a bounded linear functional of norm $1$

I want to prove that a complex-valued homomorphism $h$ on a Banach algebra $X$ which is not identically 0, is a bounded linear functional of norm $1$. This is a statement in the appendix D of the ...
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Summable family in a normed linear space

I learnt a definition: Let $X$ be a normed linear space and $J$ be a non-empty set. A family $x:J\rightarrow X$ is summable with sum $\overline{x}$ if for all $\epsilon>0$, there exists a finite ...
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Convergence of a series of vectors in a Banach space

Let $\sum_{k=1}^\infty\lambda^{k-1}\boldsymbol{v}_k$ be a series of vectors where $\lambda^{k-1}\in\mathbb{C}$, or $\lambda^{k-1}\in\mathbb{R}$, and the $\boldsymbol{v}_k$ belong to a Banach space. I ...
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1answer
12 views

Existence of adjoint operator in Euclidean space

If we define the adjoint operator of linear operator $A:E\to E$, where $E$ is a complex or real Euclidean, $n$- or $\infty$-dimensional, space, as operator $A^\ast:E\to E$ such that $\forall x,y\in ...
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1answer
15 views

Is it sufficient to check weak convergence on a (weak* or strongly) dense subset of the dual?

Let X be a Banach space. If $D \subset X^*$ is (weak*ly or strongly?) dense, then does $f(x_n) \to f(x)$ $\forall f \in D$ imply that $x_n \to x$ weakly? My thoughts: If $g_m \to g$ in the dual, then ...
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45 views

Corollary of Banach fixed-point theorem

Let $(X, \left\lVert\cdot\right\rVert)$ be a Banach space. Let $A:X\to X$ be a linear map and $\nu\in \mathbb{N}$ such that $A^k:X\to X$ is a contraction for every $k>\nu$. Is it true that for ...
2
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1answer
17 views

Solution Operator for inhomogenous Dirichlet Problem

Writing up a report, I want to understand the following solution operator better. Suppose $\Omega$ is an open bounded domain in $\mathbb{R}^d$ with boundary $\partial\Omega$ of class ...
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1answer
31 views

$\overline{\text{conv}}\{x_n: n\in \mathbb{N}\}$ has empty interior for a w-conv seqeunce $(x_n)_n$.

Given $x_n \to x$ weakly in a Banach space X, $\dim X = \infty$. Show that $\text{int}\left\{ \overline{\text{conv}}\{x_n:n\in\mathbb N \}\right\}$ is empty. Can anyone help me with this problem? ...
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25 views

C*-Algebras: Group vs. Derivation

Given a C*-algebra $\mathcal{A}$. Consider a *-derivation $\delta$. Does it always generate a group: $$\tau(t)=e^{it\delta}$$ But a group of *-automorphisms is a contraction group: ...
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1answer
21 views

CAR-Algebra: Nontriviality?

Given a Hilbert space $\mathcal{h}$. Consider the abstract CAR-algebra $a:\mathcal{h}\to\mathcal{A}_\text{CAR}$. Then their actually isometries: $$a:=a(f):\quad ...
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1answer
52 views

Hahn–Banach theorem?

In mathematics, the Hahn–Banach Theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, ...
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Proofchecking: Application of Banach-Alaoglu on weak converging nullsequence

Problem Assume $x_n \to 0$ weakly in a Banach space. Show that for all $\epsilon>0$ and for all $N\in \mathbb{N}$ there exists a $n>N$ s.t. for all $f\in X^\ast, \|f\|\leq 1$ there exists ...
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16 views

When (weakly) compact operators have pre-adjoints?

Given a bounded linear operator $T\colon X^*\to X^*$ for some Banach space $X$. Then $T$ is an adjoint of an operator $S\colon X\to X$ if and only if $T$ is weakly* to weakly* compact. Are there some ...
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28 views

Bochner Integral: Measurability

Problem Given a measure space $\Omega$ and a Banach space $E$. Consider a Bochner measurable function $S_n\to F$. Then it admits an approximation from nearly below: $$\|S_n(\omega)\|\leq ...
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1answer
100 views

Stone's Theorem Integral: Bad Example!

Disclaimer This thread is just to record. See: Answer own Question It is stated as question for jeopardy. Have fun. :) Problem Given a finite Borel measure $\mu(\mathbb{R})<\infty$ and a Banach ...
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1answer
64 views

Bochner Integral vs. Riemann Integral

Disclaimer This thread is meant to record. See: Answer own Question Anyway, it is written as problem. Have fun! :) Reference This thread is directly related to: Bochner Integral: Axioms Bochner ...
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1answer
36 views

sup norm of operator

Let $T$ be a compact linear operator defined as $$ T\circ u = \int_a^b k(x,y)\,u(y)\,dy, $$ where $k(x,y)\in C([a,b]\times[a,b])$ and $k(x,y)\ge0$ for all $x,y$, and $u\in C([a,b])$. Suppose that the ...
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Dichotomy for global existence or blow up for solutions of evolution problems.

Consider the problem (Nonlinear Schrödinger equation) \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u\mp u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ ...