# Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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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**

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**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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**

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**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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 ...

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votes

**0**answers

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

**0**answers

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

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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

**1**answer

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} ^{\...

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votes

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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**

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**0**answers

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

**0**answers

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 $\...

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vote

**0**answers

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

**0**answers

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

**0**answers

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 ...

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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**

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**0**answers

97 views

### Green function of a Laplace operator in an annulus

How to find the Green function of Dirichlet laplacian in an annulus?

**46**

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**5**answers

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 ...

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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

**1**answer

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)| \...

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**0**answers

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**

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**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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 ...