Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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Follow up on star algebra (proof verification)

I previously asked this question about a proof of the following claim: If $A$ is a commutative non-unital non-zero $C^\ast$ algebra then $\Omega (A)$ is not empty. In the meantime I believe to ...
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1answer
23 views

Sobolev's inequality from Reed & Simon vol. II

In Reed & Simon vol. II, an inequality called Sobolev's inequality is stated in Eq. (IX.19): Let $0<\lambda<n$ and suppose that $f\in L^p(R^n)$, $h \in L^r(R^n)$ with $p^{-1} + r^{-1} + ...
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1answer
90 views

Irrational Rotation

Let $\sigma$ be a homeomorphism of $S^1$. Then the following statements are equivalent; (1) O(z) is dense in S for some z in S, (2) O(z) is dense in $S^1$ for every z in $S^1$, (3) $\sigma$ is ...
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1answer
326 views

Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to ...
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31 views

Banach algebra with empty character space. [closed]

What's an example of an abelian non-zero Banach algebra such that $\Omega (A)$, the character space of $A$, is empty?
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1answer
108 views

The weak topology of a product

Let $E$ and $F$ be normed vector spaces and let $E^{\sigma}$, resp. $F^{\sigma}$ be $E$, resp. $F$ with the weak topology associated with the elements of the duals $E^*$, resp. $F^*$. Then, for ...
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1answer
161 views

If $h$ is closer to $f$ than to $g$, its integral on $\{f > g\}$ must “agree” with $f$'s?

I have the following question (a result I would like to prove, with an admirable record of failing at it so far) -- I actually do not know if it is obvious or just plain wrong. Let $f,g,h\colon ...
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27 views

Functional characterization of zeroth law of thermodynamics [Sepration of Variables]

Zeroth law of thermodynamics is stated also as: If A is in thermal equilibrium with B and if B is in thermal equilibrium with C, then A is in thermal equilibrium with C. This can be formulated ...
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1answer
48 views

$l_{\infty}$ is the quotient of $l_{1}(\aleph_{1})$?

I wonder if $l_{\infty}$ is the quotient of $l_{1}(\aleph_{1})$. If so, how to prove it?
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1answer
36 views

Unconditional bases equivallent to permutations of basis elements.

On page 9 of The Handbook of the Geometry of Banach Spaces: Volume I I found the following: "A basis $\{x_n\}_{n=1}^{\infty}$ is said to be an unconditional basis provided that $\sum \alpha_n x_n$ ...
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16 views

Weyl Operators: Discontinuity

Let $\mathcal{A}_{CCR}(\mathcal{H})$ be a CCR algebra over a Hilbert space $\mathcal{H}$. Then the Weyl operators are unitary and therefore $\sigma(W(f))\subseteq \mathbb{S}$ so by the spectral ...
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111 views

If $u_n \to u$ in $C([0,T];H^{-1})$ and $\lVert u_n\rVert_{L^\infty} \leq C$ then $u_n(t) \rightharpoonup u(t)$ in $L^1(\Omega)$?

Let $u_n \to u$ strongly in $C([0,T];H^{-1}(\Omega))$ and suppose that $u_n$ is uniformly bounded in $L^\infty((0,T)\times\Omega)$. Then $$u_n(t) \to u(t)$$ weakly in $L^1(\Omega)$ for each $t$? Can ...
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46 views

Understanding the Euler operator

While reading this book I came across a differential equation $$t^5\frac{d^2y}{dt^2}+2t^4\frac{dy}{dt}-y=0$$ that was then rewritten in terms of the Euler operator, $\delta=t\frac{d}{dt}$, with the ...
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2answers
197 views

Left topological zero-divisors in Banach algebras.

Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by $$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$ where $ ...
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1answer
144 views

Looking for different proofs of “Discrete Liouville's Theorem”.

Good day. There is a question I have already encountered twice, in very different contexts, that is relatively simple looking, but both solutions I know involve some pretty advanced theorems from the ...
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1answer
30 views

Why is $\Gamma (A)$ closed

Let $A$ be a commutative $C^\ast$ algebra and let $\Gamma: A \to C_0(\Omega (A))$ be the map $a \mapsto \widehat{a}$. Here $\Omega(A)$ denotes the character space of $A$. Why is $\Gamma (A)$ closed ...
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25 views

Another question about $C^\ast$ algebra

If $A$ is a Banach algebra then let $\Omega (A)$ denote the character space of $A$. Apparently there exist abelian Banach algebras such that $\Omega (A) = \varnothing$. Also apparently, if $A$ is a ...
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56 views

How to prove this map is injective

Let $A$ be a non-unital $C^\ast$ algebra and let $M(A)$ denote the multiplier algebra and let $\widetilde{A}$ denote the unitisation of $A$. Consider the map $\varphi : \widetilde{A}\to M(A)$ ...
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96 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
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47 views

Space of continuous functions vanishing at infinity

Let's denote with $C_0(X)$ the space of continuous functions $f$ on $X$ such that for every $\epsilon>0$ there exists a compact set $K_\epsilon\subset X$ satisfying $sup_{x\notin K_\epsilon ...
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24 views

Dirac delta function and well behaved function [duplicate]

whether dirac delta function a well behaved function? Can u please explain the properties of a well behaved function..?
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1answer
22 views

Check complex differentiability

I am trying to take a derivative w.r.t $z\in\mathbb{C}$ of the following map: $z\mapsto \sum_{j=0}^{\infty}\lambda_{j} (T(\psi+zh))_{j}$ where $(\lambda_{j})$ is a bounded sequence, $T$ is a ...
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1answer
90 views

On an estimate of sequences with weights

Does there exist a $C > 0$ such that $$ \sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4} $$ for all $a_n \geq 0$ with ...
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1answer
38 views

Adjoint of sum of two operators

Let $A$ be self-adjoint and $B$ symmetric (which means densely defined for me as well) with $A$-bound less than $1$. Does this imply that $(A+iB)^*=A-iB$ ?
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61 views

Find vector space $X$; so that vector space operations are not continuous

How to choose $X$ to be a complex vector space with a topology $\tau$ on $X$; so that vector space operations are not continuous with respect to $\tau$; that is, the mappings, $X\times X \to X: ...
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2answers
31 views

Ultra weakly closed *-subalgebra of B(H)

I'm currently working on a text about von Neumann algebras and the author used without further clarifying that any ultra weakly closed *-subalgebra of $B(H)$ contains a largest projection. Could ...
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28 views

Differentiate a log of $L^p$ norm, don't understand this result

I'm reading this paper. In it, the authors show this lemma: And then they prove this lemma My question is: I have no idea how they get the result in Lemma 3.2. Do we not get $$\frac{d}{ds}\log ...
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1answer
14 views

How to show $\mu_A(x)=\inf\{\alpha>0: \alpha^{-1} x\in A\}$ where $\mu_A(x)$ is the Minkowski functional of $A$?

Let $X$ be a $\mathbb K$-vector space ($\mathbb K=\mathbb R$ or $\mathbb C$) and suppose $A\subset X$ is convex (and absorbing). How to show $$\{x\in X: \mu_A(x)<1\}\subset A?$$ Above ...
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90 views

When does a real function together with its derivatives form a basis?

Consider real functions defined on an interval, let's say $[-\pi,\pi]$. Let $s(x)$ be an analytic, periodic function such that all of its Fourier components are non-zero. We build a set of functions ...
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39 views

Topology of Normed Space

$(X, \lVert \cdot \rVert)$ is a normed space. Let $x \in X \setminus \{0\}$ and $Y \subset X$ is a subspace. Prove that if $Y$ is open then $Y=X$. Which technique is more useful? We know ...
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1answer
28 views

Operator Tensor Product

Let $S$ and $T$ be bounded operators over a Hilbert space $\mathcal{H}$. Define their tensor product $S\otimes T$ as acting on $\mathcal{H}\otimes\mathcal{H}$ by $S\otimes T(x\otimes y):=Sx\otimes Ty$ ...
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82 views

What is a predual of the Banach space of compact operators on $\ell^2$?

I am wondering if the space $K(\ell^2)$ of compact operators on $\ell^2$ can have a predual. Thank you in advance for your help.
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18 views

A locally convex space is metrizable if and only if it is first countable

I'm studying Functional Analysis by myself. the following is an exercise while I'm not sure about my answer. If $X$ is a locally convex space (LCS), show that $X$ is metrizable if and only if $X$ is ...
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1answer
53 views

Prove that $(C_{00},\|\cdot\|)_{\ell^2}$ is not a Banach space

How can we prove that $(C_{00},\|\cdot\|)_{\ell^2}$ is not a Banach space? How can we find counter example for this problem?
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29 views

Unbounded Operators: Notation?

For continuous a.k.a bounded operators we have $\mathcal{B}(X,Y)$ stressing on boundedness and $\mathcal{L}(X,Y)$ stressing on linearity entailing $\mathcal{C}(X,Y)$. Is there a notation for ...
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812 views

Reflexive space which is not uniformly convex

I found this beautiful theorem (Milman-Pettis): Every uniformly convex Banach space is reflexive I think it's a remarkable statement, since uniformly convexity is a geometric property of the norm ...
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1answer
25 views

Extend the Stone-Weierstrass theorem to high dimension?

I am thinking of if there is high dimensional extension to the well known Stone-Weirstrass theorem. Wikipedia says it is possible to extend the 1D theorem to 2D, i.e. If  f  is a continuous ...
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1answer
40 views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
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1answer
25 views

Every unitary representation of a compact group is a direct sum of irreducible representations.

I've read nice proofs of a few different variants of the Peter-Weyl theorem and its corollaries. For instance I know that for $G$ a compact group, $L^2(G)$ is a Hilbert space direct sum of the matrix ...
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1answer
67 views

Any idea with this problem of distances???

Let $E$ a normed linear space and $H$ the closed hyperplane $H=\ker f$, where $f\in L(E,\mathbb{R})$, $f\not\equiv 0$. Show that if $a\in E$ then $$d(a,H)=\frac{|f(a)|}{||f||}$$ And the problem have a ...
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1answer
57 views

How can I prove that $f$ is continuous at $0$?

Let $E$ and $F$ linear normed spaces, and consider a linear function $f:E\to F$ which satisfies for all $(x_n)\in E^{\mathbb{N}}$ such that $x_n\to 0$ then $f(x_n)$ is bounded in $F$. Then I have to ...
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1answer
20 views

Showing $\int_{-1}^1\int_{-1}^1(u_x^2+2u_y^2+u^2-x^2y^2u)\, dx\, dy\geq c$.

Prove that for some $c\in\mathbb{R}$: $$G(u) =\int_{-1}^1\int_{-1}^1(u_x^2+2u_y^2+u^2-x^2y^2u)\, dx\, dy\geq c$$ for every $u \in H_0^1$. I know that $$G(u) ...
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1answer
25 views

Pseudospectrum of an $n\times n$ matrix has at most n connected components

For $\epsilon>0$, the $\epsilon$ pseudospectrum of an $n\times n$ matrix $A$ is given by $\sigma_{\epsilon}(A)=\{z\in \mathbb{C}:\|(z-A)^{-1}\|>\epsilon^{-1}\}$, with the convention that ...
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1answer
23 views

Dual cone is a cone

Let $E$ be a Banach space and $P$ be a cone. A nonempty convex closed set $P\subset E$ is called a cone if it satisfies the conditions: $x\in P,$ $\lambda \geq 0$ implies $\lambda x\in P$; $x\in P,$ ...
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1answer
26 views

Polar decomposition in a von Neumann algebra

Let $M \subseteq B(H)$ be a von Neumann algebra and $T \in M$. If $T=U|T|$ is the polar decomposition of T, why is $U \in M$? I'm thinking it's because $M$ is SOT-closed, but I'm not entirely sure.
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1answer
20 views

Projections in Tensor Product

Let $X$ and $Y$ be Banach spaces (algebras). If $X\otimes^\pi Y$ denotes the projective tensor product of $X$and $Y$, define $P:X\otimes^\pi Y\rightarrow X$ as follows : for $x\otimes y\in X\otimes ...
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3answers
44 views

Reference on $\mathcal{L}^p(I;X)$

I am doing some reading on evolution equations, and $\mathcal{L}^p$ spaces with functions with values in a Banach space $X$ appears rather often. However I have not found a comprehensive reference ...
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21 views

a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
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1answer
23 views

Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentialky selfadjoint if it contains a dense subset of analytic vectors?
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1answer
21 views

Monotone log-supermodular function is supermodular.

Let $X$ and $Y$ be lattices. Let $f: X \times Y \rightarrow \Re$. Function $f$ is log-supermodular if for all $x'>x$ and $y'> y$ \begin{equation} f\left(x', y'\right)f\left(x, y\right) \geq ...