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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).

32
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0answers
566 views

Hahn-Banach From Systems of Linear Equations

In this paper1 on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables $c_i = A_{ij} x_j$: \begin{...
28
votes
0answers
783 views

Why are functions with vanishing normal derivative dense in smooth functions?

Question Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm? Here I define $...
22
votes
0answers
323 views

$A$ 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 $...
20
votes
0answers
732 views

Fast convergence in $L^1$ implies convergence almost everywhere

This is a proof-verification request. Claim: Let $(X,\mathscr M,\mu)$ be a measure space. Let $f_n$ ($n\in\mathbb N$) and $f$ be measurable, integrable, real-valued functions such that $(f_n)_{n\in\...
19
votes
0answers
319 views

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 ...
18
votes
0answers
480 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 ...
18
votes
0answers
486 views

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 ...
17
votes
0answers
783 views

Two possible definitions of “vector-valued distribution”

Let $X$ be a reflexive Banach space. Define $$\tag{1} \mathcal{D}^\star(0, T; X)=\left\{ u\colon \mathcal{D}(0, T)\to X\ \text{linear and continuous}\right\} $$ where the topology on the space of ...
15
votes
0answers
1k views

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 ...
14
votes
0answers
281 views

If $V\subset L^\infty[0,1]$ with $\|f\|_\infty \leq c\|f\|_2$, then $V$ is finite dimensional

If $V$ is a linear subspace of $L^\infty[0,1]$ with $\|f\|_\infty \leq c\|f\|_2$ for all $f\in V$, then $V$ is finite dimensional. The proof is an explicit calculation: Since $L^\infty[0,1] \subset L^...
14
votes
0answers
4k views

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 $\...
14
votes
0answers
628 views

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 ...
13
votes
0answers
224 views

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 ...
13
votes
0answers
680 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 ...
13
votes
0answers
193 views

Topological Space in which every compact subset is metrizable

Is there an (more or less) established name for the class of topological spaces in which every compact subset is metrizable? This is true for example in (LF)-spaces (inductive limits of Frechet-spaces)...
12
votes
0answers
251 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 \...
12
votes
0answers
447 views

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^{...
12
votes
0answers
808 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: $$\frac{d}{dt}\...
12
votes
0answers
1k views

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" ...
12
votes
0answers
551 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 ...
12
votes
0answers
1k views

$L^p$ 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 ...
11
votes
0answers
222 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 ...
11
votes
0answers
212 views

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-...
11
votes
0answers
480 views

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: ...
11
votes
0answers
616 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 $L^p$, ...
10
votes
0answers
168 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 ...
10
votes
0answers
189 views

Can (linear) differential equations of infinite order be recast into equations of first order?

In most analysis courses one sees that differential equations of order $n$ are basically a subset of higher dimensional differential equations of order $1$, for example the equation: $$f^{(n)}(t)=F\...
10
votes
0answers
139 views

How to get the idea of the formula for the mean value property for the heat equation

From the mean-value property of the Laplace's equation, we have the following mean-value property: $$ u(x)=\frac{1}{a(n)r^n}\int_{B(x,r)}u\,dy. $$ But for the mean-value property of the Heat equation, ...
10
votes
0answers
244 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 ...
10
votes
0answers
770 views

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 $n$ ...
9
votes
0answers
283 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 ...
9
votes
0answers
105 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 (...
9
votes
0answers
249 views

Lifting spectral gap to covering space

Let $M$ a complete Riemannian manifold. It is well known that the Laplace-Beltrami operator on $M$ is essentially self-adjoint and thus has a unique self-adjoint extension $\Delta_M$ in $L^2(M)$. The ...
9
votes
0answers
146 views

Isomorphic matrix algebras with non-isomorphic C*-algebras

Let $A$ and $B$ be two $C^{\ast}$-algebras which their matrix algebras, $M_2(A)$ and $M_2(B)$, are $\ast$-isomorphic $C^\ast$-algebras. Question 1: Are $A$ and $B$ isomorphic $C^\ast$-algebras? In a ...
9
votes
0answers
285 views

Green's functions/fundamental solution to a non-constant coefficients pde

We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta ...
9
votes
0answers
164 views

Homological categories in functional analysis

I got the feeling that some of the "usual categories" in functional analysis could be homological (though, excuse my ignorance, I don't know anything about functional analysis, yet). E.g. in "Lectures ...
9
votes
0answers
399 views

Heat equation proving smoothness

I have a question regarding a PDE course: Let $T$ be the strongly continuous semigroup which belongs to the heat equation, thus with generator $A$ is the Laplacian. Suppose we have $g \in C^{\infty}...
9
votes
0answers
525 views

locally convex space generated by a countable family of seminorms is metrizable

Let $X$ be a locally convex Hausdorff topological space, whose topology if generated by the countable family of seminorms $\{p_i:\space i\in\mathbb{N}\}$. I'd like to prove that $X$ is metrizable. So, ...
9
votes
0answers
192 views

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 ...
9
votes
0answers
730 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}{...
8
votes
0answers
132 views

Is there a notion of a continuous basis of a Banach space?

If $X$ is a Banach space, then a Hamel basis of $X$ is a subset $B$ of $X$ such that every element of $X$ can be written uniquely as a linear combination of elements of $B$. And a Schauder basis of $...
8
votes
0answers
77 views

Space of linear, continuous, hyperbolic functions is open, dense in the set of invertible functions

Let $(X,||\cdot||)$ be a Banach space on $\mathbb{C}$ and $\mathcal{L}(X)$ the set of linear, continuous functions from $X$ to itself. For $T\in\mathcal{L}(X)$, define the norm $||T||_{\mathcal{L}(X)}:...
8
votes
0answers
78 views

Eigenvector Riesz basis under operator multiplication?

I recently encountered the Riesz Spectral Operators which roughly speaking are closed operators whose eigenvectors form a Riesz basis and I became interested in when such operators can be perturbed ...
8
votes
0answers
179 views

The sum of eigenvalues of integral operator $S(f)(x)=\int_{\mathcal{X}} k(x,y)f(y)d\mu(y)$ is given by $\int_{\mathcal{X}} k(x,x) d\mu(x)$?

Setup: Let $(\mathcal{X},d_{\mathcal{X}})$ and $(\mathcal{Y},d_{\mathcal{Y}})$ be two separable metric spaces. Let $M^1(\mathcal{X})$ be the space of Borel probability measures on $\mathcal{X}$ with ...
8
votes
0answers
931 views

The Weak topology on an infinite-dimensional space is not metrizable

Let $X $ be an infinite-dimensional normed space. I want to prove that the weak topology on $X$ is not metrizable. This is my solution: Assume that there is a metric $d$ on $X$ inducing the weak ...
8
votes
0answers
166 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
8
votes
0answers
460 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
8
votes
0answers
379 views

Riesz's 1909 proof of the Riesz Representation Theorem

Frigyes Riesz originally proved the Riesz Representation Theorem on $ C[0,1] $ -- here is his 1909 paper in English (original French). He builds a real valued function $ \text{A} $ on $ [0,1] $ ...
8
votes
0answers
188 views

Sobolev space on $M \times [0,\infty)$, $M$ compact closed manifold

Consider a manifold of the form $M \times [0,\infty)$, where $M$ is a closed compact manifold without boundary. So $M \times [0,\infty)$ is a semi-infinite cylinder. I want to know information about ...
8
votes
0answers
234 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 $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...