# Questions tagged [fa.functional-analysis]

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

9,257
questions

2
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32
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### When a compact subset of a TVS can be continuously projected on a closed linear subspace?

Let $V$ be a (Hausdorff) topological vector space, $W\subset V$ a closed linear subspace, $X\subset V $ a compact.
Q: When there is a continuous map $P:X\to W$ such that $P(x)=x$ for every $x\in X\...

1
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0
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58
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### Is the Schwartz space a tame Frechet space?

I ran into the following definition of tame Frechet spaces and Nash-Moser therem.
It says that the space of smooth functions on a compact manifold is tame Frechet.
However, I wonder if
The Schwartz ...

1
vote

1
answer

123
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### How is interpolation used in the proof of Lemma 4.1 in Tao's article Endpoint Strichartz Estimates?

In the proof of Lemma 4.1, pp. 962–963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some ...

6
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1
answer

289
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### Do acyclic amenable groups exist?

Is there an example of a nontrivial discrete amenable group with vanishing integral homology?
To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...

0
votes

1
answer

149
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### Does weak convergence in $L^2$ imply convergence a.e. of a subsequence?

The title pretty much explains it all. Let $u_n\in L^2(\mathbb{R}^n)$ be a sequence converging weakly in $L^2$ to some $u\in L^2(\mathbb{R}^n)$, that is $\int u_k v \to \int u v$ for all $v\in L^2(\...

1
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0
answers

43
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### Convergence of slice in an equivalent renorming

Let us consider $\ell_2$ space with $\Vert \cdot \Vert_2$ norm. Let us define a new norm equivalent to $\Vert \cdot \Vert_2$ norm as follows:
$$
\Vert x \Vert_0 = \max \{ \Vert x \Vert_2, \sqrt{2} \...

2
votes

1
answer

117
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### Are projective tensor products left-exact if one considers only maps of norm at most 1?

Consider the category $\mathrm{Ban}$ of Banach spaces and bounded linear maps and the category $\mathrm{Ban}_1$ of Banach spaces and bounded linear maps of operator norm at most 1. Let $\otimes_\pi$ ...

0
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3
answers

243
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### A generalisation of Tchebychev inequality

Let $f,g \in C(\mathbb R)$ with $\exists M \in \mathbb R^*, \forall (x,y) \in \mathbb R^2, M\times (f(x)-f(y))(g(x)-g(y)) \geq 0$.
Is it true that exists $ u$ any real function, and $a,b$ monotone ...

1
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1
answer

69
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### $L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?

Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...

2
votes

1
answer

220
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### Decompose a function into a bounded part and a Lipschitz part

Let $f: \mathbb R^d \to \mathbb R^d$ be a measurable function such that
$$
\sup_{x,y \in \mathbb R^d} \frac{|f(x) - f(y)|}{\max \{1, |x-y| \}} < \infty.
$$
Are there functions $g,h: \mathbb R^d \...

2
votes

2
answers

161
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### $L^p$ domination of mixed partial derivatives by the unmixed ones?

Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...

1
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0
answers

46
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### Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE

Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in
H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of
the ...

2
votes

0
answers

55
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### Constructing spaces of analytic sections. Silva spaces

This is a follow up on this question Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations to add more details.
In the linked question I ...

8
votes

1
answer

255
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### Is the dual of a Fréchet space weakly* separable?

It is known that if $X$ is a separable Banach space with dual $X^\ast$, then $B_{X^\ast}$, the closed the unit ball in $X^\ast$, is compact and metrizable in the weak* or $\sigma(X^\ast, X)$-...

2
votes

1
answer

142
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### Are these two norms on localized versions of $L^p_q$ equivalent?

$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$.
Let $E$ be the space of all real-valued ...

1
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0
answers

68
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### positive invertible maps which are not *-automorphisms

Let $A$ be a unital C*-algebra. Is there a unital positive self-map $F:A\to A$ which is invertible (i.e. injective and surjective) but not a $*$-automorphism? If yes, how does appear its Gelfand-...

5
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2
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685
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### How can I prove this special version of the Poincaré inequality?

I am reading a paper, Is $L^2$ Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?, which uses an "application" of Poincaré's inequality. The form I know and ...

3
votes

1
answer

249
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### Schroedinger operator in 2 dimensions with singular potential

Consider the Schroedinger operator
$$H = -\Delta + \frac{c}{\vert x \vert^2}$$
in two dimensions with $c >0$
This operator has a self-adjoint realization, since it is a positive symmetric operator ...

0
votes

1
answer

111
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### Singular integral bounded by Dirichlet form?

We define for some fixed $L$
$$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$
in particular $x_1,x_2 \in \mathbb R^2.$
Let $f \in C_c^{\infty}(\Omega)$, then I am ...

2
votes

1
answer

124
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### Difference in essential spectrum between Schrodinger operators

I am considering two Schrodinger operators on $\mathbb{Z}^2$ and compare their essential spectrum. The operators are both of the form $H=A+V$ where $A$ is the adjacency operator on the $\mathbb{Z}^2$-...

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0
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35
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### Convergence of function sequences derived from systems of linear equations in operations research

I am working on a problem in operations research that involves systems of linear equations with a specific structure, and I am seeking insights or references that might help in proving a pointwise ...

2
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0
answers

59
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### Commutant of rank one operators over Hilbert module

Let $\mathcal{E}$ be a right Hilbert $C^*$-module over the $C^*$-algebra $A$. Consider for $\xi, \eta\in \mathcal{E}$ the rank-one operator
$$\theta_{\xi, \eta}: \mathcal{E}\to \mathcal{E}: \zeta \...

3
votes

1
answer

127
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### Rescaling Fourier coefficients of a continuous function by a bounded sequence

This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...

2
votes

1
answer

91
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### $L^2$ space of Hilbert-Schmidt operator valued functions

Let $\mathscr{S}$ denote the space of all Hilbert-Schmidt operators on $L^2(\mathbb R)$. Consider the Hilbert space $L^2(\mathbb R, \mathscr S)$ of square-integrable $\mathscr S$-valued function, that ...

0
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0
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55
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### asymptotic expansions for $C^{1+\epsilon}$operators

I want to understand the asymptotic expansion for $C^{1+\epsilon}$ operators.
More precisely, if a complex operator $T(z)=\sum_{n \ge 1}T_n z^n$ is defined on a closed unit disk, and it is $C^{1+\...

1
vote

1
answer

70
views

### Convexity property of an equivalent norm on $\ell_2$

Let us consider the space $\ell_2$ with an equivalent norm defined by
$$
\Vert x \Vert = \max \{ \Vert x^{'} \Vert_2, \Vert x^{''} \Vert_2 \},
$$
where $x^{'}=(0, x_2, x_3, \cdots)$, $x^{''} = (x_1, 0,...

6
votes

2
answers

232
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### If a Banach / Fréchet manifold $M$ happens to be a topological vector space, is $M$ just a Banach / Fréchet space?

In finite dimensions, if $M$ is a smooth manifold that happens to be a vector space, then it is indeed just the Euclidean space.
I wonder if the same result holds valid in infinite dimensions. More ...

1
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0
answers

55
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### Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?

$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$.
Let us consider a Schwartz space $\mathcal{S}$ defined as
\begin{equation}
\mathcal{S}:= \Bigl\{ \...

1
vote

1
answer

89
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### Image of a complete topological group under open and surjective map is complete?

A uniform space $X$ is complete if every Cauchy filter in $X$ is convergent. Here we do not require $X$ to be Hausdorff.
Question.
Let $G$ be a complete topological group and let $H$ be a topological ...

1
vote

1
answer

92
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### Maximal element w.r.t. abolute continuity of measures

Suppose that $\mu$ is a $\sigma$-finite measure on $\mathcal{X}\equiv\bigotimes_{i=1}^n\mathcal{X}_i$. Let $\Pi$ denote the set of all $\sigma$-finite product measures on $\mathcal{X}$. Define
$$
\...

6
votes

0
answers

191
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### Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized

Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...

0
votes

0
answers

72
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### Ultraproduct reflexive

Hello I have the following construction: Let $(E,\|\cdot\|_E)$ be a Banach space, $E_n:=E^n$ and $\|x_n\|_n:=\frac{1}{n}\sum_{k=1}^n \|x_n(k)\|_E$ for all $x_n=(x_n(1),...,x_n(n))\in E^n$ and $n\in\...

2
votes

1
answer

135
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### Invertibility of message passing with invertible parametrization

Consider the message passing framework defined by, $$f(\boldsymbol{x}_i)= \boldsymbol{x}_i + \sum_{j \neq i} (\boldsymbol{x}_i -\boldsymbol{x}_j) g(\|\boldsymbol{x}_i -\boldsymbol{x}_j\|^2),$$ for $i=...

10
votes

1
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226
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### Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?

In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$.
The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...

2
votes

0
answers

121
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### Convergence of eigenfunctions

In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...

3
votes

1
answer

76
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### Can gradient zero implies that a function is constant with Hörmander vector fields

Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by
$$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...

1
vote

1
answer

117
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### Points in the Stone Cech compactification are intersection of open sets

Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...

4
votes

0
answers

56
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### Find reasonable definition for endpoint Lorentz function spaces $L^{\infty,q}$ via the idea from endpoint Triebel-Lizorkin ${\scr F}_{\infty,q}^s$

On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...

3
votes

1
answer

163
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### Can homomorphisms be Borel in some weaker topology?

Let $X$ be a Banach space with separable dual and let $A$ be a Banach algebra. Consider a norm continuous homomorphism $h$ from $L(X)$, the Banach algebra of bounded operators on $X$ onto $A$. In $L(X)...

2
votes

0
answers

97
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### Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$

Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...

-1
votes

0
answers

28
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### Conditional joint distribution decomposition

Let $I$ be a finite index set and $\mathcal X_i$ for $i\in I$ be a standard measurable space (or say a Polish space or $\mathbb R$). Suppose that the probability measure $P(X_I)$ on the product space $...

3
votes

1
answer

95
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### Dimension of spectral projection subspaces under local convergence

I'm interested in estimates on dimension of spectral projection subspaces of some limit operator. I recently asked a related question in the thread Dimension of spectral projection subspaces under ...

2
votes

1
answer

154
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### Does there exists an example of a Banach space that is compactly LUR; but not LUR

We know that a Banach space $X$ is locally uniformly rotund(LUR) if for $x, x_n \in S_X$ with $\Vert x+x_n \Vert \to 2$, we have $x_n \to x$. In the same case, if $(x_n)$ has a convergent subsequence, ...

0
votes

0
answers

49
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### Behaviour of duality mapping in a uniformly convex and smooth Banach space

Let us first define the normalized duality mapping $J:B \to 2^{B^*}$ by
$$
J(\xi)= \{ jx \in B^* : \langle jx , \xi \rangle = \Vert jx \Vert_{B^*}^2 = \Vert \xi \Vert_B^2 \}, \forall \xi \in B,
$$ ...

1
vote

0
answers

106
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### Building representation of an arbitrary umbral calculus

Consider a set of integrable functions on the interval $(0,1)$.
Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function).
In such system the ...

0
votes

0
answers

208
views

### Gauss transformation in fractional Sobolev space

Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that
$$
\int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...

3
votes

2
answers

144
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### Dimension of spectral projection subspaces under strong convergence of operators

I have a possibly simple question regarding estimating bounds on spectral projection subspace.
Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$ converging in the ...

5
votes

0
answers

159
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### Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions

Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such ...

-1
votes

0
answers

52
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### How this second ODE is solved by Bessel’s function

Any help how this equation ( Eq. B3 in paper ) :
$$
\phi’’(\eta) + \frac{6(1+\omega)}{1+3\omega)} \frac{1}{\eta} \phi’(\eta) + \omega k^2 \phi(\eta) =0
$$
Has been solved by Bessel’s function with a ...

4
votes

0
answers

99
views

### Weakly null sequences in projective tensor products II

The question in this post is the question below from an article by Rodriguez & Rueda Zoca [1].
Below is a complimentary salad/side dish that accompanies the main course.
Let $B^2(X,Y)$ denote ...