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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Haar's base for $L^2[0,1]$

$\newcommand{\span}{\operatorname{span}}$ Define $e_{0,0}\equiv 1$, and for all $n\in \mathbb{N}$ $$e_{n,k}=\begin{cases} 2^{n/2} &\text{if } \frac{k-1}{2^n}\leq x\lt \frac{k-\frac{1}{2}}{2^n}\\ ...
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863 views

Compact subset of an open set

Let $X$ be complete separable metric space and $A\subset X$ is open. Does it mean that there is a compact subset of $A$? My solution is the following: since $A$ is open there is $B(x,r)\subset A$, ...
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Dual of a dual cone

Any hint on how to prove the following please: Let $K$ be a convex cone, and $K^*$ its dual cone. Prove that $K^{**}$ is the closure of $K$. Thanks!
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471 views

Is compactness a stronger form of continuity?

Let $H$ be a Hilbert space. We say that a linear operator $T \colon H \to H$ is compact if it maps bounded sets to precompact ones, that is, if for every bounded sequence $(a_n)$ in $H$, $(Ta_n)$ has ...
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149 views

Two terms that I want to understand: weakest topology and jointly continuous (in the following context).

I was reading an article online, please help me to understand the following lines (in bold letters). - Topological structure: If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and ...
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186 views

Basic question about $\sup_{x\neq 0}{} \frac{\|Ax\|}{\|x\|} = \sup_{\|x\| = 1}{\|Ax\|} $, $x \in\mathbb{R}^n$

I am having trouble with understanding the following definition while studying some basic things related with matrix norms: For every matrix $A\in M_n(\mathbb{R})$ $$\sup_{x\neq 0}{} ...
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294 views

Is $\left(\sum\limits_{k=1}^\infty \frac{x_k}{j+k}\right)_{j\geq 1}\in\ell_2$ true if $(x_k)_{k\geq 1}\in\ell_2$

Let $(x_k)_{k\geq 1}\in\ell_2$. Consider $\left(\sum\limits_{k=1}^\infty \dfrac{x_k}{j+k}\right)_{j\geq 1}$. Now my question is that whether ...
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177 views

Orthonormal Sets in Hilbert Spaces

let $H$ be a Hilbert space and let M be a dense linear subspace of H. Can we find a complete orthonormal set $\{u_{\alpha}: \alpha \in A\}$ for H in M? I think the answer is negative in general, but ...
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298 views

Introductory/Intuitive Functional Analysis Book

Can you recommend a gentle introduction to the abstract thinking and motivation of functional analysis? I'm looking for a book that holds you by the hand and shows the details of exercises, etc. ...
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293 views

Selfadjoint compact operator with finite trace

I have a compact selfadjoint operator $T$ on a separable Hilbert space. For some fixed orthonormal basis, the operator's diagonal is in $\ell^1(\mathbb{N})$. Can we conclude that $T$ is trace ...
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629 views

Showing a subset of $C([0,1])$ is compact.

Let $${\cal F}=\left\{ f:\left[0,1\right]\to\mathbb{R} : \left|f\left(x\right)-f\left(y\right)\right|\le\left|x-y\right|\mbox{ and }{\displaystyle \int_{0}^{1}f\left(x\right)dx=1}\right\}.$$ Show ...
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240 views

Does the Banach algebra $L^1(\mathbb{R})$ have zero divisors?

Assume that the functions $f,g: \mathbb{R}\rightarrow \mathbb{R}$ are integrable and equal to zero on $(-\infty,0)$, (i.e $f,g \in L^+$). Then by Titchmarsh's theorem: $f*g$ is zero almost everywhere ...
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On Some Properties of Hölder Continuous Functions

The function space $H^{\alpha} (\Omega)$ for $0 < \alpha \le 1$, is the set of functions: $$\{ f \in C^0(\Omega) : \sup_{x \neq y} \dfrac{|f(x) - f(y)|}{|x-y|^{\alpha}} < \infty \}$$ with the ...
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323 views

Why is every positive linear map between $C^*$-algebras bounded?

We know that every positive linear functional on a $C^*$-algebra is bounded. How can we prove every positive linear map between $C^*$-algebras is bounded?
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528 views

Weak-* sequential compactness and separability

Let $X$ be a Banach space, and let $B$ be the closed unit ball of $X^*$, equipped with the weak-* topology. Alaoglu's theorem says that $B$ is compact. If $X$ is separable, then $B$ is metrizable, ...
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236 views

reference for operator algebra

I am taking a course on operator algebra this semester. My instructor has suggested a reference "Kadinson and Ringrose." Are there any other good/standard references for this subject that I can look ...
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229 views

Compact maps problem in Lax

In Functional Analysis of Peter Lax there are the following exercise Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$. ...
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1answer
132 views

Criterion for convergence of the sequence of powers of a linear operator to $0$

Let $T$ be a linear operator in a Banach space $\mathbf{B}$. Suppose that for every $x \in \mathbf{B}$ there exists some real numbers $c_x>0$ and $a_x<1$ such that $||T^nx|| \leq ca^n$, for all ...
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1answer
318 views

Linear contraction on a Banach space

Let $X$ be a Banach space with a norm $\|\cdot\|_1$ and $A$ be a linear operator on $X$ such that $\|A\|_1\leq 1$; $\|A^m\|_1<1$ for some $m\in \mathbb N$. Is that true that there is an ...
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Properties of dual spaces of sequence spaces

Can you tell me if I got the following homework right? Nitpicking is welcome. a) Recall that $$ c_0 (\mathbb{N}) = \{ f: \mathbb{N} \rightarrow \mathbb{C} \mid \lim_{n \rightarrow \infty } f(n) ...
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246 views

$X\subsetneqq Y$ but $X^\star=Y^\star$

Are there $X,Y$ real Banach spaces, such that $X\subsetneqq Y$ (strictly contained) and $X^\star=Y^\star$, where $\star$ denotes the topological dual? This property is not true for Hilbert spaces, ...
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424 views

$C_0(X)$ is not the dual of a complete normed space

Let $X$ be any locally compact Hausdorff space and assume that it is not compact. I've heard that the Banach space $(C_0(X),\|\!\cdot\!\|_\infty)$ is not isometrically isomorphic to the (norm) dual of ...
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292 views

pointwise limit on a complete metric space

Let $\{f_n: X\rightarrow \mathbb{R}\}$ be a sequence of continuous real-valued functions on a complete metric space, $X$. Suppose this sequence has a pointwise limit, $f$. How easy is it to see that ...
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Norm on a Hölder's space

I want to prove that Hölder space is a Banach space under the "Hölder Norm" ie. $\|\cdot\|_{C^{k,\alpha}}$. Any hints would be appreciable .
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$T$ is continuous if and only if $\ker T$ is closed

Let $X,Y$ be normed linear spaces. Let $T: X\to Y$ be linear. If $X$ is finite dimensional, show that $T$ is continuous. If $Y$ is finite dimensional, show that $T$ is continuous if and only if $\ker ...
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489 views

$\ell_\infty$ a Grothendieck space

The problem I am considering stated formally is this: Show that if a sequence in $\ell_\infty^*$ is weak*-convergent, then it is also weakly convergent. We may reduce this to the case where the ...
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719 views

Completeness of a finite direct sum of closed subspaces of $L^2$

Let $X_1$ and $X_2$ be real-valued square-integrable random variables defined on a probability space $(\Omega, {\cal F},P)$. For $i=1,2$, set $$ A_i := \{g(X_i)\in L^2 \mid g \text{ is some Borel ...
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On the limits of weakly convergent subsequences

Let $\{ f_n \}$ be a sequence in a Hilbert space $L^2(\mathbb{R}^d)$. We say that this sequence converges weakly to an element $f \in L^2$ if $\langle f_n, g \rangle \to \langle f,g \rangle$ for every ...
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What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
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603 views

Eigenfunctions of the Laplacian

I am willing to offer a bounty for this one, so I will give you an exact idea of what I need: I am looking for solutions of $$\Delta \Psi(r,\theta)=k^2\Psi(r,\theta)$$ where $k\in \mathbb{R}$. Such ...
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217 views

A commutator identity for bounded linear maps and the identity operator of a non-zero normed space is never a commutator

Let $ \mathcal{X} $ be a normed linear space and $ S,T: \mathcal{X} \to \mathcal{X} $ be linear operators such that $ S \circ T- T \circ S=1 $. Show that $ S \circ T^{n+1}- T^{n+1} \circ S=(n+1)T^n ...
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288 views

A convergence problem in Banach spaces related to ergodic theory

Suppose $X$ is a Banach space, $T\in B(X)$, satisfied the following condition. $\sup \Big\lVert\frac{1}{n}\sum \limits_{i=0}^{n-1}T^{i}\Big\rVert<\infty$ $\frac{1}{n}\lVert ...
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1answer
270 views

An application of J.-L. Lion's Lemma

Let $X,Y$ and $Z$ be three Banach spaces with norms $\|\cdot\|_X$, $\|\cdot\|_Y$,$\|\cdot\|_Z$. Assume that $X\subset Y$ with compact "injection" and that $Y\subset Z$ with continuous injection. Then ...
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192 views

Monotone convergence to a fixpoint in a Banach space

Let $\mathscr X$ be a complete separable metric space and $\mathbb B$ be the Banach space of all real-valued bounded measurable functions on $\mathscr X$. The partial order on this space is introduced ...
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Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
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218 views

Definition of Banach limit

In my Bachelor Thesis I have defined a Banach limit as a functional $LIM: l^\infty (\mathbb{N})\rightarrow \mathbb R$ that has the following properties: B1 If $(x_n)$ is a convergent sequence, then ...
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172 views

Traces on separable simple $C^{\ast}$- algebras

What is an example of a separable, simple $C^{\ast}$-algebra that admits two different tracial states? EDIT: Julien has pointed to a number of avenues to answer this question. If anyone has an ...
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466 views

Characterization of weak convergence in $\ell_\infty$

Is there some simple characterization of weak convergence of sequences in the space $\ell_\infty$? If yes, is there some similar claim for nets? I was only able to come up with a characterization ...
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1answer
415 views

Liouville's theorem for Banach spaces without the Hahn-Banach theorem?

Let $B$ be a (complex) Banach space. A function $f : \mathbb{C} \to B$ is holomorphic if $\lim_{w \to z} \frac{f(w) - f(z)}{w - z}$ exists for all $z$, just as in the ordinary case where $B = ...
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1answer
252 views

Strong topology vs Natural topology

Let $X$ be a locally convex space and $\left< X, X^{\prime} \right>$ stands for the dual pair. The bidual of $X$ is denoted by $X^{\prime \prime}$ and this is a dual of $X^{\prime}$ with a ...
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142 views

Is $W_0^{1,p}(\Omega)$ complemented in $W^{1,p}(\Omega)$?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $p\in (1,\infty)$. It is know that there exist a unique bounded surjective linear map $T: W^{1,p}(\Omega)\to W^{1-1/p,p}(\partial\Omega)$ with ...
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96 views

Is there a smooth compact set between any two compact sets?

today I saw the following statement and of course believe that this is true but I don't know how to prove it rigorously (and neither do my colleagues). Let $\Omega \subset \mathbb{R}^n$ be open and ...
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1answer
224 views

Practical applications of the $L^p$ norm when $p \neq 1,2,\infty$

I'm roughly familiar with the concept of $L^p$ norms -- what they represent and how they are computed -- though I am far from educated in functional analysis in general. For reasons that are more or ...
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361 views

Locally integrable functions

Formulation: Let $v\in L^1_\text{loc}(\mathbb{R}^3)$ and $f \in H^1(\mathbb{R}^3)$ such that \begin{equation} \int f^2 v_+ = \int f^2 v_- = +\infty. \end{equation} Here, $v_- = \max(0,-f)$, $v_+ = ...
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1answer
191 views

Abelian sub-C*-algebras

Given a non-abelian C*-algebra $A$. I am wondering what are the possible abelian sub-C*-algebras of $A$. Let $K$ be the spectrum of $A$. Does $A$ contain an isomorphic copy (as a Banach space) of the ...
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147 views

Maximum of measures over sets and functions

Let $(X,\mathcal A)$ be any measurable space and denote by $\mathrm b\mathcal A_1$ the set of all real-valued $\mathcal A$-measurable functions $f$ satisfying $\|f\|:=\sup_{x\in X}|f(x)|\leq 1$. Let ...
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84 views

*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$

I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form $A\in ...
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Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space

I have to show that the real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space. Certainly, I have to search for a cauchy sequence of real numbers ...
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325 views

Integer translates of a scaling function

I think this is asked as a standard exercise in books about wavelets (e.g. exercise 7.2 in Mallat's book), but I couldn't find a proof. Let $\phi$ be a scaling function (see definition below). I ...
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1answer
134 views

Calculus on the Sobolev space valued function of one real variable $t$?

Now I am interested in the calculus on Banach space valued function, especially the function with value in a certain Sobolev space. I want to prove that $$\bigcap_{k=0}^m ...