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Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then $\...
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39 votes
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$A$ and $B$ commute on a dense set but $e^{iA}$ and $e^{iB}$ do not

Let $A$ and $B$ be unbounded, symmetric operators on a Hilbert space $H$ with a common domain $D$. If $AB = BA$ on $D$, is it necessarily that case that $e^{iA}$ and $e^{iB}$ also commute? If $A$ and $...
user15464's user avatar
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32 votes
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Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
Torsten Schoeneberg's user avatar
30 votes
1 answer
835 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 ...
Ben W's user avatar
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29 votes
1 answer
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Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
timofei's user avatar
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27 votes
1 answer
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Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex ...
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24 votes
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Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
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21 votes
1 answer
2k views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
Ufuk Can Bicici's user avatar
18 votes
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731 views

Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Let $F\subset\Bbb{R} $ intersect every uncountable $\mathcal{F}_{\sigma}$ set. $B\subset \Bbb{R}$ is said to have the property of Baire if $B=U\triangle M$ where $U$ is open and $M$ is meager. Does ...
Sourav Ghosh's user avatar
18 votes
1 answer
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If a map between separable Banach spaces has closed graph, does it have a point of continuity?

It is well known that the closed graph theorem does not directly extend to nonlinear maps: even for functions from $\mathbb{R}$ to $\mathbb{R}$, having closed graph does not imply continuity. But let'...
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17 votes
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Is $\frac{1}{H(x) \pm i0}$ a distribution if $|\nabla H| \neq 0$ for $H(x)=0$?

I know that $\frac{1}{x \pm i0}$ is a tempered distribution in $\mathcal{S}'(\mathbb{R})$, see e.g. the Sokhotski–Plemelj theorem. In some lecture notes online I found the following statement (without ...
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17 votes
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Nuclear spaces vs Banach spaces

The Wikipedia article on nuclear spaces say the following: "There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: ...
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On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
blindman's user avatar
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$L^p$ spaces with negative $p$

Let $p$ be a negative real number and $X$ a measure space. Define $L^p(X)$ to be the vector space of all measurable functions $X \to \mathbb{C}$. Then define a "norm" on this space as is usual in the ...
Dejan Govc's user avatar
16 votes
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Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
anon's user avatar
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15 votes
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Are the unconditionally convergent series, with terms in a Banach algebra, closed under the Cauchy product?

We have a Banach algebra $\mathbb L$, and two sequences $(A_0,A_1,A_2,\cdots),\;(B_0,B_1,B_2,\cdots)\in\mathbb L^{\mathbb N}$, for which the sums $\sum_{n\in\mathbb N}A_n,\;\sum_{n\in\mathbb N}B_n$ ...
mr_e_man's user avatar
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15 votes
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How to construct examples of functions in the Spaces of type $\mathcal{S}$

There are $3$ $\mathcal{S}$-type Spaces, namely $\mathcal{S}_\alpha\:,\: \mathcal{S}^\beta\:,\:\mathcal{S}_\alpha^\beta$. They are defined by: $\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le C_qA^kk^{...
creative's user avatar
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15 votes
1 answer
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Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
fbg's user avatar
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15 votes
0 answers
634 views

Finding a proper solution of a given functional

It's my first post here, but I worked very hard to find solution and I failed. Hereinafter, I skip physical background and directly proceed to my mathematical problem. No matter how, you know the ...
Miloslav Capek's user avatar
14 votes
0 answers
743 views

Computing the total variation for a multivariable function

I am trying to write an example computation with multivariable total variation to include in my functional analysis notes using the following definition from Wikipedia: Let $\Omega$ be an open subset ...
WDR's user avatar
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Existence of function satisfying $f(f'(x))=x$ almost everywhere

My project is to Study the existence of a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ differentiable almost everywhere satisfying $ f\circ f'(x)=x$ almost everywhere $x \in \mathbb{R}$...
Pascal's user avatar
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13 votes
1 answer
364 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
Rajesh D's user avatar
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358 views

Compact embedding in weighted Sobolev spaces

I have a question concerning Sobolev's embedding. Let (for simplicity) $\Omega=\left( 0,1\right) $. Then it is well known by Rellich's theorem that $H^{1}\left( \Omega\right) $ is compactly ...
Anton Raute's user avatar
12 votes
0 answers
366 views

Non-trivial faces of the closed convex hull of a non-convex closed set with connected complement

I'm trying to prove or disprove a problem, but I'm struggling to make headway. Any help is appreciated. Suppose $X$ is a Hilbert Space, and $C \subseteq X$ is closed, bounded, non-convex, and $X \...
Theo Bendit's user avatar
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12 votes
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300 views

What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
Cameron Williams's user avatar
12 votes
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Question about boundedness of a sequence in $ W^{3,q} $ for any $ 1\leq q < \frac{N}{N-1} $

I have asked this question several months ago, I have understood every thing and there are good comments and they have helped me , but only I have a question about tomas comment, how can Calderón-...
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12 votes
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Dual of $\ell^p$ Direct sum

I am asked to show that the $\ell^p$-direct sum of a sequence of Banach Spaces $X_n$ is isometrically isomorphic to the $\ell^q$ direct sum of $X_n^*$ where $X_n^*$ is the dual of $X_n$ for each $n$ ...
Jack's user avatar
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0 answers
738 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
michek's user avatar
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12 votes
0 answers
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Eigenprojection as Contour Integral over Resolvent

Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
Meneldur's user avatar
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11 votes
0 answers
359 views

Extremely rigorous (research level) treatment of the laplace transform

Edit: I have found what I needed in Schwartz's Mathematics for the Physical Sciences. Will type up a reply when I have time. Bourbaki does not explain the justifications behind operational calculus ...
Qqqq123123's user avatar
11 votes
0 answers
200 views

A (possible) generic spectral property in one dimensional dynamics

This question was previously posted on MathOverflow. Context and Definitions Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [MvS]) if: $T$ has a finite ...
Matheus Manzatto's user avatar
11 votes
0 answers
274 views

What is the interpretation/intuition of $e^{itA}$ for a self-adjoint unbounded operator?

Let $A : i \frac{d}{dt} : D(A) \to H^1([0,1])$ with domain $D(A)=H^1_*([0,1])=\{u \in H^1([0,1]): u(0)=u(1)\} \subseteq H^1([0,1])$. Then I know that $A$ is self-adjoint. Using the spectral theorem, ...
Suspicious Fred's user avatar
11 votes
0 answers
302 views

A projection operator is linear iff $X$ is a Hilbert space

This question comes from Linear and Nonlinear Functional Analysis with Applications (Philippe G. Ciarlet), Chapter 4, Problem 4.3-4. 4.3-4 Let $\mathcal{P}_n[0,1]=\left\{\left.p\right|_{[0,1]} ; p \...
Hang's user avatar
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11 votes
0 answers
579 views

Understanding Lang's Proof of Fubini's Theorem

This question concerns the proof of Theorem 8.4 (Fubini's Theorem part 1) on page 162 in Lang's real and functional analysis book. To understand the proof I need to give following background from the ...
Alphie's user avatar
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11 votes
0 answers
426 views

Potential for Monotone Operator

I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The authors claim to construct a convex ...
Pete Caradonna's user avatar
11 votes
0 answers
278 views

A characterization of the "Direct Integral" construction in terms of the properties it satisfies?

Fortunately there's a wonderful thing called the "direct integral" which enables one to make sense of direct sums of uncountably infinite families of Hilbert spaces. Unfortunately I've tried to read ...
Saal Hardali's user avatar
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11 votes
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Are uncountable "Schauder-like" bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
Martin Sleziak's user avatar
11 votes
0 answers
305 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
yess's user avatar
  • 1,002
11 votes
0 answers
350 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
Indigo's user avatar
  • 554
11 votes
0 answers
6k views

Is $L^p$ separable?

Whether a $L^p(X,\mu)$ space is separable? I understand that the answer depends on $p$ and $X$. It seems to me that it is separable when $1\leq p < \infty, X=\mathbb{R}^n$ or $X=\mathbb{N}$. ...
Zach's user avatar
  • 306
11 votes
0 answers
1k views

Hilbert transform and Hilbert matrix

The Hilbert matrix is \begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\[4pt] \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & & \ddots \\[4pt] \frac{1}{...
plm's user avatar
  • 1,515
11 votes
0 answers
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Gradient operator the adjoint of (minus) divergence operator?

Recently I found this statement -- the gradient operator is the adjoint of the minus divergence operator -- in one of my lecture notes. Knowing only a little about functional analysis, I'm looking for ...
Ailurus's user avatar
  • 1,192
11 votes
0 answers
2k views

Fixed point: linear operators

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ ...
SBF's user avatar
  • 36.1k
10 votes
0 answers
216 views

Does a distribution in the dual space of $C_0^\infty$ extend continuously to $C_b^\infty$?

I want to understand distributions that are bounded linear functionals on smooth functions whose all derivatives vanish at infinity. Before asking questions, let me first define the terms used in this ...
user141240's user avatar
10 votes
0 answers
397 views

The Heat Equation in Brezis' book

I am reading the heat equation in Functional Analysis, Sobolev Spaces, and Partial Differential Equations by Haim Brezis, and having some concerns about the proof, whose screenshot is as attached ...
Justin Lien's user avatar
10 votes
0 answers
190 views

Linear optimization for functions

I have the following linear optimization problem. $$ \max \int_0^1 w(t) dt $$ subject to $$ \int_0^1 w(t) \, x_i(t) \, dt \geq 0, \quad i=1,\dots,n $$ and $$ 0 \leq w(t) \leq 1 \quad \text{for all} \...
Bogdan's user avatar
  • 351
10 votes
0 answers
99 views

New norm with strictly coarser induced topolgy

Let $(V,\|\cdot\|)$ be an infinite dimensional normed space. Does there alway exist a norm $|||\cdot|||$ on $V$ which induces a strictly coarser topology than $\|\cdot\|$? I know, that there is ...
Claire's user avatar
  • 4,384
10 votes
1 answer
177 views

Nonlinear funtionals of smooth maps between Riemannian manifolds

Let two smooth Riemannian manifolds $M$ and $N$, and let $C^{\infty}(M, N)$ be the family of smooth maps between them. I would like to study functionals of the form $$E[\cdot]: C^{\infty}(M, N) \to \...
them's user avatar
  • 1,042
10 votes
0 answers
685 views

Generalized limits

Cross-posted to Mathoverflow. $\DeclareMathOperator{\Lim}{Lim}$ $\DeclareMathOperator{\dom}{dom}$ $\DeclareMathOperator{\shift}{\sigma}$ $\DeclareMathOperator{\cesaro}{C}$ After reading Terry Tao's ...
user76284's user avatar
  • 5,977
10 votes
0 answers
163 views

Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

$\newcommand{\Tr}{\operatorname{Tr}}$Let $\mathcal{S}(\mathbb{R}^k)$ denote the $k$-dimensional Schwartz space with the usual topology, and let $\mathcal{S}'(\mathbb{R}^{k}))$ denote its strong dual (...
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