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: Isometry

Setting Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with unit $1\in\mathcal{A}$. Consider a morphism: $\pi:\mathcal{A}\to\mathcal{B}$ without $\pi[1]=1\in\mathcal{B}$. Especially, it is a ...
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C*-Algebras: Distinct Spectra

Given unital C*-algebras $1\in\mathcal{A}$ and $1'\in\mathcal{A}'$. Regard an element $A\in\mathcal{A}\cap\mathcal{A}'$. Can it happen that: $\sigma(A;\mathcal{A})\neq\sigma(A;\mathcal{A}')$ ...
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32 views

Open mapping lemma - are these versions equivalent?

Here is a version the Open Mapping Lemma given in class : Let $X$ be a Banach space and $Y$ be a normed space. Let $T : X\rightarrow Y$ be a bounded linear map. Assume there exist $M \geq 0$ and ...
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C*-Algebras: Contractive Morphism

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with $\mathbb{1}_\mathcal{A}\in\mathcal{A}$. Consider an algebraic morphism $\pi:\mathcal{D}\subseteq\mathcal{A}\to\mathcal{B}$ with ...
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32 views

Show that a subspace is closed in Hilbert space $H$

Let $u\in B(H)$ , $\lambda < 0$. Also we have $\|(u-\lambda)x\|\geq |\lambda|\|x\|$. So $u-\lambda$ is bounded below. To show $(u-\lambda)(H)$ is closed in $H$, suppose $\{(u-\lambda)x_n\}$ be ...
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36 views

Prove that $T^*$ is injective iff $ImT$ Is dense

Let X,Y be two normed spaces, and $T:X\rightarrow Y$ a bounded linear operator. prove that the adjoint operator $T^*$ ($T^*f(x)=f(Tx)$ is injective iff $ImT$ is dense any help would be great guys. I ...
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1answer
45 views

Proving existence of a linear functional

Let $(X, \| \cdot \|)$ be a normed space, and let $A, B ⊂ X$ be disjoint convex sets such that $B$ is closed and $A$ is compact. Prove that there exists $\varphi ∈ X^*$ such that $$\sup_{a\in A} ...
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Existence of a mapping in a nonseparable Banach space that moves all nearby points to far-away points

Does there exist a nonseparable Banach space $X$, a mapping $F: X\to X$, and an open nonempty subset $D\subset X$ such that $$ \forall\,E>0 \quad \exists\,\delta>0: \quad \forall\,x,y\in D \quad ...
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35 views

Show that an operator is well-defined

Let $v\in B(H)$, Define $u:|v|H\to H$ such that $u(|v|\xi) = v\xi$ . To show the map $u$ is well-defined, the author writes $$\||v|\xi\|^2=\langle v^*v\xi,\xi\rangle = \|v\xi\|^2$$ But I do not know ...
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19 views

Partial isometry and projection

The following is a Theorem of Murphy's C*-algebras and operator theory: Let $H_1, H_2$ be Hilbert spaces and $u\in B(H_1,H_2)$. If $u^*u$ is a projection, then $uu^*u=u$. To show it, for $\xi\in ...
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44 views

Equality of two operators

The following is a fact in Murphy's C*-algebras and operator theory page 49: Suppose $u,v \in B(H)$, where $H$ is a Hilbert space, then $u=v$ if and only if $\langle u\xi,\xi\rangle = \langle ...
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1answer
58 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
63 views

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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Alternate proof of a result on dual spaces: what is wrong with it?

I am familiar with Rudin's book's proof of the fact that, in $\sigma$-finite measure spaces and for $p\in[1,+\infty)$, the dual space of $L^p$ is $L^q$ where $p,q$ are conjugate, i.e. ...
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33 views

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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1answer
161 views

Parabolic PDE $\to$ ODE on Banach space

Would someone please explain to me the concept of converting a parabolic PDE to an ODE on Banach space? If I have a PDE, say $$u_t = f(u_{xx}, u_x, u, p)$$ where $p$ is a parameter and the solution ...
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18 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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25 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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63 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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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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1answer
24 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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30 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
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The example of a mapping is homomorphic but not isomorphic

The example of two Banach space A,B such that the mapping T:A→B, is onto , one-to-one and homomorphic but not isomorphic i.e ∥T(x)∥≠∥x∥. I think there are two norm spaces, such that norm does not ...
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28 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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54 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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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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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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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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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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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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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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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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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
36 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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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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Banach Algebra spectral theory [closed]

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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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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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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0answers
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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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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About Banach Spaces And Absolute Convergence Of Series

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
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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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1answer
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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
45 views

Banach Spaces: Pointwise Limit vs. Uniform Limit

Agreement All notions are up to null sets. Limits are meant by simple functions. Problem Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. Consider bounded measurable ...
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1answer
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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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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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136 views

Generalized Riemann Integral: Bounded Nonexample?

Reference For a convergence theorem on integral see: Riemann Integral: Uniform Convergence For an improper version of integral see: Riemann Integral: Improper Version For a comparison of integrals ...
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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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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 ...