Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
2
votes
0answers
91 views
Chern number of projection-Topological magic in physics
I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...
2
votes
0answers
55 views
Compact Kaehler submanifolds of projectivized Hilbert space
If we take a separable complex Hilbert space $H$, its projective space $PH$ is an infinite-dimensional Kähler manifold in a fairly obvious sense (see below). Suppose $M \subset PH$ is a finite-...
0
votes
1answer
163 views
Stone–von Neumann theorem?
The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...
0
votes
1answer
150 views
Uniformly Bounded
If $a_1<1$, $a_1+a_2+a_3>1$, for $x,y,z>0,$
(1) define a fucntion
$$H(x,y,z)=\frac{x^{\frac{1}{2}}\int_0^{\infty}\frac{1}{t^{a_1} (1+t)^{a_2+1}
(1+t+z)^{a_3}}\exp\big\{-\frac{x}{1+t}-\...
-4
votes
0answers
92 views
Le produit de série entier [on hold]
Salam. J’aurais une question concernant les séries entiers, j’aurais besoin de votre aide pour ceci résulta de cette multiplication:
$$t(∑_{n=0}^{∞}a_{n}t^n)(∑_{n=0}^{∞}(a_{n}t^n)-a)(∑_{n=0}^{∞}(a_{n}...
4
votes
1answer
158 views
Pushing Cuckoo Eggs under Inverse Radon Transforms
Essentially the inverse of the Radon transforms $Rf(L)=\int_L{f(x)|dx|}$ has the ability to reconstruct $f(x)$ from the integrals over all lines; or, expressed differently to (re)construct $f(x)$ ...
1
vote
1answer
95 views
Poisson equation on noncompact manifold
Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold and $C^\infty_c$-functions are dense in $L^p_k$ space.
For the equation
$$\Delta u=f,$$
...
2
votes
2answers
123 views
center of a $C^*$-algebra
Does there exist a $C^*$-algebra $A$ such that the center of $A$ is $0$ and $A$ also has a tracial state?
I know the fact that the center of $\mathcal{K}(H)$ is $0$, but $\mathcal{K}(H)$ has no ...
1
vote
1answer
63 views
Interpolation of a trilinear functional
Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If ...
4
votes
0answers
39 views
Interpolation of some Sobolev spaces
Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$
I am wondering what is
$$(X_0,X_2)_{1/2,2}=?$$
Would it be $H^2_0(0,...
4
votes
0answers
75 views
Generalized Gelfand Triples
Normally, when we talk about Gelfand triple we have three Hilbert spaces
$$\newcommand{\X}{\mathcal{X}}
\X_+ \subset \X_0 \subset \X_-
$$
such that the subsets are dense, the embedding mappings are ...
0
votes
0answers
83 views
Why is $\widetilde{W}$ closed?
let's consider $\mathscr{U}$ a free ultrafilter on the natural numbers and consider its corresponding ultrapower
\begin{align*}
\widetilde{X} = (\ell ^{\infty}(X _{i})/\operatorname{ker}\mathcal{N},\...
1
vote
1answer
141 views
Approximation of a two-variable function by tensor products
Let $X$ and $Y$ be compact metric spaces and $f: X \times Y \to \mathbb{R}$ be a continuous function.
We know that, for every $n \in \mathbb{N}$, by the Stone-Weierstrass theorem, there exist $k_n \...
7
votes
1answer
152 views
Almost orthonormal projection and orthonormal projection in Hilbert space
Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e.
$$\langle e_i, e_j \rangle \approx \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$
and $\alpha$ is ...
0
votes
0answers
77 views
On the infimium of a functional
Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$
where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...
1
vote
1answer
68 views
The difference between the nonlocal and local conditions problems
In some problems involving ordinary differential equations, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed.
In this paper: Existence and uniqueness ...
-2
votes
0answers
58 views
Spectrum and operators [closed]
Hwo can help me to prove that if the spectrum of a normal operator lies on a circle {z∈C:∣z∣=1}, then this operator is unitary.
I'm very thankful for any ideas and help
4
votes
0answers
109 views
Weighted reverse Poincare inequality over a function class of neural networks
We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
0
votes
0answers
24 views
equivalent definition of k-rank numerical range of an operator
Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...
4
votes
2answers
135 views
Density of Lacunary Functions
I'm curious whether lacunary functions (functions in the complex plane that are holomorphic in some open ball about the origin, but cannot be analytically extended past that ball) are typical or the ...
3
votes
1answer
96 views
What “mild solution” means, and how to find it?
In this paper: Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem Byszewski studied this form of functional-differential nonlocal problem (1)...
0
votes
0answers
33 views
Anisotropic elliptic problems
Where can I find a reference on the following anisotropic problem?
$$(\bullet) \qquad \begin{align*} \nabla_x (A(x) \nabla_x u(x,y)) + \Delta_y(K(y) \ast u(x,y))\end{align*}=0, $$
where $(x,y) \in \...
0
votes
1answer
70 views
The definiton of a multiplier on a Banach algebra
Let $A$ be a Banach algebra. Some textbooks define a (left ) multiplier as a map $T:A\rightarrow A$ satisfying $T(ab)=T(a)b$ for all $a,b\in A$ and assume that $A$ needs to be a without order Banach ...
3
votes
1answer
105 views
Lower estimate of the minimal eigenvalue of a Hamiltonian
Consider a linear operator $H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ given by
$$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$
where $V\colon \mathbb{R}^3\to \mathbb{R}$ is a continuous (or ...
1
vote
1answer
57 views
Elliptic interface problem without conditions on the interface
Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...
0
votes
0answers
69 views
If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$
Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\...
2
votes
0answers
82 views
Limit circle/point of an ODE with finite eigenvalues
Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
4
votes
0answers
63 views
Is the Baireness a 3-space property of topological groups
It is known that the product of two Baire spaces can be meager.
On the other hand, by a recent result of Li and Zsilinszky the product of two Baire spaces is Baire if one of the spaces is countably ...
0
votes
1answer
100 views
$B _{\ell ^{2}} ^{+}$ with the norm $\lVert\lvert \cdot \rvert\rVert _{\sqrt{2}}$ doesn't have normal structure
$\newcommand\binorm[1]{\lVert#1\rVert}\newcommand\trinorm[1]{\lVert\lvert#1\rvert\rVert}$Consider the space $\ell ^{2}$ with the standard norm
\begin{align*}
\binorm x_{2} = \left( \sum _{i =1} ^{\...
-4
votes
0answers
126 views
Finding a paper [closed]
Could you please help me to find the paper "Rüssmann, Helmut Kleine Nenner. I. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes. (German) Nachr. Akad. Wiss. Göttingen Math.-Phys. ...
4
votes
0answers
82 views
How to think about dual space of a certain space of Lipschitz functions
Consider the following Banach space (for concreteness):
$$X=Lip(\bar{\mathbb{B}}^n)=\{f\in C^0(\bar{\mathbb{B}}^n): \Vert f \Vert_L<\infty \}$$
where
$$
\bar{\mathbb{B}}^n=\{\mathbf{x}\in \mathbb{...
6
votes
0answers
89 views
Homomorphisms from BV
Denote by $\mathsf{BV}(\mathbb T)$ the Banach space of functions on the circle with bounded variation which is a Banach algebra under the pointwise product. Is there a surjective homomorphism from $\...
1
vote
0answers
78 views
Limit of sequence of vectors in $\ell^2$ with coefficients approaching $0$
Let $\{v_m\}_{m \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over the complex plane $\mathbb{C}$ such that: $\{v_m\}_{m \in \mathbb{N}}$ is linearly independend and $v_m \to v$
Let $V= \...
3
votes
0answers
43 views
Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”
I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...
2
votes
0answers
46 views
Do the gradient of convex (Fenchel) conjugates preserve the “distance” between two uniformly convex functions?
Update: I have discontinued pursuing this question. However, by observing that the conjugate and its derivative are nothing more than optimum and optimizer, my question should be answered by carefully ...
10
votes
0answers
186 views
Are biduals of spaces of differentiable functions on hypercubes Grothendieck?
Consider the space $E_n = C^1([0,1]^n)$ of continuously differentiable functions with the usual norm
$$\max\{ \|f\|_\infty, \|f^\prime_{x_1}\|_\infty, \ldots, \|f^\prime_{x_n}\|_\infty\}.$$ making it ...
0
votes
0answers
97 views
Green function of a Laplace operator in an annulus
How to find the Green function of Dirichlet laplacian in an annulus?
46
votes
5answers
4k views
Jean Bourgain's Relatively Lesser Known Significant Contributions
A great mathematician is no longer with us.
Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions, such as the proof of Vinogradov's conjecture ...
0
votes
0answers
54 views
Prove that an iterative estimate implies Holder continuity
Let $u$, $w$ be nonnegative continuous functions such that $\frac{u}{w}$ is bounded on $B_{2^{-1}}$. Why the inequality
$$a_k \le \frac{u}{w} \le b_k \quad \text{ on $B_{2^{-k}}$} , \qquad b_k - a_k ...
4
votes
1answer
129 views
Flat norm metrizes the weak* topology
I've come across the following statement in literature (without proof or reference) about the flat norm of currents
$$
F(T) = \sup \{ T(\omega) : \omega \in D^k(U), |\omega(x)| \leq 1, |d\omega(x)| \...
0
votes
0answers
84 views
Holder-Sobolev type inequality
Let $U$ be a bounded subset of $\mathbb{R}^n$. Let $p>n$. Let $W^{2,1}(U_T)$ be the Banach space of functions $u:U\times[0,T]\rightarrow\mathbb{R}$ with the norm $\|u\|_{W^{2,1}(U_T)}=\sum_{2s+|\...
0
votes
1answer
192 views
Does this norm have a specific name? Banach space? References?
Let $(X,\mathscr{B},\mu)$ be a $\sigma$-finite measure space. Let $\gamma$ be a probability measure on $L_2(\mu)$ with $\mathrm{supp} \, \gamma = L_2(\mu)$ and existing first moment. Then
$$
f \...
2
votes
1answer
61 views
Uniform inequality for an analytic perturbation
Let $T$ be a bounded linear operator acting on a complex Banach space. Suppose that $T$ has spectral radius strictly less than $1$. If we introduce an analytic perturbation to $T$, $s\mapsto T_s$ for $...
3
votes
0answers
347 views
What tools from functional analysis are relevant to investigating this operator?
Given a sequence of continuous functions ${{f_n}}$, define the varicontinuity index $$V({f_n}): \mathbb{R} \to [0, \infty]$$ by
\begin{split}
V({f_n})(x) &=\sup \Big\{\varepsilon > 0\big|\; \...
0
votes
0answers
33 views
Properties of the dual optimizer in Wasserstein loss
Let $P=\frac 1 k \sum_i N(\theta_i, \sigma^2 I)$ and $Q=\frac 1 k \sum_i N(\mu_i, \sigma^2 I)$ be two mixtures of Gaussians in $\mathbb{R}^d$. A well known fact is that the Wasserstein-$2$ loss ...
5
votes
1answer
158 views
Complemented subspaces constructed from finite pieces- part II
This is a follow up to: Complemented subspace constructed from finite pieces
Suppose $Y=\overline{\cup E_n}$ is a closed subspace of a separable Banach space X, where each $E_n$ is a $n$-dimensional ...
0
votes
1answer
68 views
Complemented subspace constructed from finite pieces
Suppose $Y=\overline{\cup E_n}$ is a closed subspace of a Banach space, where each $E_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E_n\subseteq E_{n+1}$. Can one ...
2
votes
0answers
35 views
Integral equation with kernel defined in a rectangle
Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds=0} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$
Observe that the kernel is not defined on a square.
My question: Can I apply the classical ...
6
votes
1answer
126 views
If $\ $ $yx_n\to 0 $ for all $y$ in a C$^*$-algebra, Is it true that $x_n$ is weakly convergent to $0$?
$A$ is a C$^*\! $-algebra and $(x_n)_{n\in \mathbb{N}} \subseteq A $.
If $\ $ $yx_n\to 0 $ for all $y\in A$, Is it true that $x_n$ is weakly
convergent to $0$ ?
For unitals this is trivial. ...
4
votes
0answers
65 views
Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel
Let $\Omega$ be an open subset of $\mathbb{R}^d$. Under regularity conditions, we know that the $s$-th order Sobolev space $H^s(\Omega)$ with $s \geq d/2$ is a reproducing kernel Hilbert space. In ...
