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, ...

learn more… | top users | synonyms (1)

55
votes
0answers
2k views

Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...
11
votes
0answers
770 views

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 ...
10
votes
0answers
137 views

Regularity of PDEs, general question (Example: Kolmogorov equation)

Let us consider a differential operator $L$ and the PDE $$Lu=0$$ with some boundary conditions (assume Neumann conditions). For example, Kolmogorov's equation $$(1) \quad Lu(t,x) = \partial_t u(t,x) + ...
10
votes
0answers
400 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
264 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 ...
10
votes
0answers
540 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" ...
10
votes
0answers
1k views

Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) dt$.

I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator $$K(g) = \int_0^1 e^{xt} g(t) dt.$$ The sources I've checked* seem to say that the process is fairly ...
9
votes
0answers
107 views

Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? ...
9
votes
0answers
458 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 ...
8
votes
0answers
334 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 ...
8
votes
0answers
133 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 ...
8
votes
0answers
135 views

What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
8
votes
0answers
189 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 ...
8
votes
0answers
150 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 ...
8
votes
0answers
220 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 ...
8
votes
0answers
179 views

Sampling theorem.

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, ...
8
votes
0answers
246 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
8
votes
0answers
113 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 ...
8
votes
0answers
2k views

Prerequisites for functional analysis

I'm having a few months of free time, and I decided to do a self-study on functional analysis (in-depth) in the meantime. I'm aware that functional analysis requires a good deal of foundation from ...
7
votes
0answers
87 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 ...
7
votes
0answers
62 views

What's the intuition behind the direct integral of a family of Hilbert spaces?

In order to understand better the mathematically rigorous version of Dirac's formalism in Quantum Mechanics I've been reading about direct integrals of Hilbert spaces, projector-valued measures and so ...
7
votes
0answers
119 views

When does analytic in the operator norm imply analytic in the trace class norm?

Consider $U$ a nice compact region in $\mathbb{C}$ with boundary $\Gamma$. Let $S_1$ b the ideal of trace class operators on a separable complex Hilbert space $H$. We will let $\|\cdot \|$ be the ...
7
votes
0answers
183 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 ...
7
votes
0answers
179 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] $ ...
7
votes
0answers
207 views

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 ...
7
votes
0answers
141 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: ...
7
votes
0answers
78 views

Mankiewicz theorem

I'm looking for a proof of Mankiewicz theorem, which states that: If $U, V$ are open, connected subsets of normed spaces $E, F$ respectively, then every bijective isometry $U \rightarrow V$ extends ...
7
votes
0answers
2k views

Proof of separability of Lp spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof. Questions: It says 'it is easy to construct a function $f_{2} ...
7
votes
0answers
215 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
7
votes
0answers
348 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let ...
7
votes
0answers
480 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 ...
7
votes
0answers
337 views

Show that the function is constant

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$ ...
7
votes
0answers
2k views

Are vague convergence and weak convergence of measures both weak* convergence?

For quite a long time, I have been confused about the definitions of weak convergence and vague convergence of measures among other modes of convergence that root from functional analysis, mainly due ...
7
votes
0answers
207 views

Closure in the Space of Probability Measures with the Prohorov metric

I have seen this result stated countless times: assume the metric space $(\theta,d)$ is separable; then $(\theta,d)$ is complete if and only if the space $(\mathcal{P}(\Theta),\rho)$ (the space of ...
7
votes
0answers
454 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 $X$ ...
6
votes
0answers
159 views

Brezis Exercise 3.27 extension.

Let $E$ be a separable Banach space with norm $\|\cdot\|$. The dual norm on $E^*$ is also denoted by $\|\cdot\|$. Let $(a_n) \subset B_E$ be a dense subset of $B_E$ with respect to the strong ...
6
votes
0answers
172 views

The study of the properties of the solution of Heat equation.

I am studying the following basic heat equation. (All notations follows Evans book) \begin{align} u_t -\Delta u =0 &\text{ in }\mathbb R^2\times(0,\infty)\\ u=u_0 &\text{ on }\mathbb ...
6
votes
0answers
150 views

Adjoints in Banach spaces

I been trying to figure out if the construction of the isomorphism between $g(W)$ and $f^{*}(W^{*})$ via $$B_V(v, g(w)) = B_W(f(v),w) \quad \forall v \in V, w \in W $$ described (at least this is ...
6
votes
0answers
78 views

Intuition and slick proof that distributions are special cases of hyperfunctions?

I am looking for some slick or simple proof that distributions are a special case of hyperfunctions. Furthermore, what is the intuition behind this fact? Why should one think of hyperfunctions as ...
6
votes
0answers
114 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 ...
6
votes
0answers
200 views

Regularity of a weak solution

The problem is formulated as follows: Let $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. For $u\in L_2(\Omega)$ one can define the one dimensional weak dirivative $\partial_1 u$. Now define ...
6
votes
0answers
64 views

Trace theorems for arbitrary differentiability $k$, with embedding constants under control as $k\to\infty$

The usual trace theorem (with non-optimal exponents, but I don't care for those at the moment) says that $$ W^{1,p}(\Omega)\hookrightarrow L^p(\partial\Omega) $$ for Lipschitz domains. When ...
6
votes
0answers
102 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 ...
6
votes
0answers
88 views

One more AC equivalence question

Is "Every vector space admits a norm" weaker than AC? I know that the statement follows from "Every vector space has a basis", which is equivalent to AC.
6
votes
0answers
179 views

Euler-Lagrange Equation and “Eigen Value ”

The Eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient (where $t$ is a scalar variable): $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$ ...
6
votes
0answers
132 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 ...
6
votes
0answers
102 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 $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
6
votes
0answers
129 views

The dual of the Banach space $C(\Omega)$

It is well-known that the dual of the Banach space $C([0,1])$, i.e. the space of all continuous functions on the interval, is the space of all functions of bounded variation on the interval, ...
6
votes
0answers
71 views

Two Body Schrodinger Equations

I have a question involving the eigenvalues of a two-body Schrodinger equation. Let $$H=-\frac{1}{2m}\Delta_{x_1}-\frac{1}{2m}\Delta_{x_2}+\frac{e^2}{|{{x_1}-{x_2}}|}$$ over the Hilbert space ...
6
votes
0answers
55 views

Two Hilbert spaces $V \subset H$, a basis for both spaces?

Let $V \subset H$ be a pair of Hilbert spaces (with different inner products). The embedding is continuous and dense, and both spaces are separable. Is it always the case that one can I find a ...