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Questions tagged [fa.functional-analysis]

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

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Equality of two measures on functional spaces

It is well known that if $\mu$ and $\nu$ are two measures on the space $C^0([0,1],\mathbb{R}^n)$ of continuous mappings from $[0,1]$ to $\mathbb{R^n}$ endowed with its Borel $\sigma$-algebra satisfy $$...
Anico's user avatar
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9 votes
1 answer
239 views

Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions

Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
Tian LAN's user avatar
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2 answers
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Proof that the closed convex hull of a weakly convergent sequence has empty interior using properties of $c_0$?

Let $X$ be an infinite-dimensional Banach space, and $x_n\to 0$ weakly in $X.$ Let $K$ be the closed convex hull of $\{x_n\}.$ I remember a proof that $K$ has empty interior as follows: define a map $...
Ma Joad's user avatar
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13 votes
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331 views

Are dualizable topological vector spaces finite-dimensional?

Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product. Every finite-dimensional ...
Dmitri Pavlov's user avatar
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77 views

Hilbert spaces that include algebraic polynomials

This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
FDK's user avatar
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1 vote
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86 views

Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?

Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
mathlover's user avatar
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Question on complemented subspaces of a product space

Assume that we have closed subspaces $Y_1$ and $Y_2$ of Banach spaces $X_1$ and $X_2$, respectively. If the product $Y_1\times Y_2$ is complemented in $X_1\times X_2$, does it follow that $Y_i$ is ...
Davor's user avatar
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3 votes
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118 views

Existence of very weak solution to the elliptic equation $\partial_i(a^{ij}\partial_j u)=\partial_k\partial_l f$

Let $a^{ij}\in W^{1,n}\cap L^\infty (B^1)$ be uniformly elliptic, i.e. $\lambda|\xi|^2\le a_{ij}(x)\xi_i\xi_j\le \Lambda |\xi|^2$ for a.e. $x\in B^1$, $\xi\in\mathbb R^n$, where $B_1\subset \mathbb R^...
Tian LAN's user avatar
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Extracting each field operator as Wightman fields from a set of time-ordered products satisfying Eckmann-Epstein axioms

The paper by Eckmann-Epstein proves that Schwinger functions at "coinciding points" uniquely defines "time-ordered products". In physics, these "time-ordered products" ...
Isaac's user avatar
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Multivariate polynomial approximation

Let $f$ be a function on $[-1,1]^d$ with some smoothness property, for example, it is in the Sobolev space $W^{k,p}$. Let $P_n$ is a space of polynomials with degree $n$. My question is what is the ...
Iris's user avatar
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4 votes
2 answers
310 views

Injectivity of a convolution operator

Let $p,\mu,\nu$ be probability density functions on $\mathbb{R}$ such that $$ \int_{\mathbb{R}}p(y-x) \nu(y) \, dy=\mu(x). $$ Now, consider the operator $T:L^2(\mu)\to L^2(\nu)$ such that $$ Tf=f*p.$$ ...
Ribhu's user avatar
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How to prove that $f(x) := |x|^{\frac{\lambda - n}{p}}(1 - \psi(x))$ satisfies a specific property related to its limit at the origin

Disclaimer. I have asked this question a month ago on MSE (click here to access the original post) and even bountied it. I got an answer on MSE, but unfortunately I don't feel like it has enough ...
Rodrigo's user avatar
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2 votes
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Continuity of disintegrations in non locally compact spaces

Let $X$ and $Y$ be Radon spaces, $\mu$ a Borel probability measure on $X$, $F\colon X\to Y$ measurable. Then the disintegration theorem gives us a disintegration $\{\mu^y\}_{y\in Y}$ of $\mu$ with ...
Nathaël's user avatar
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196 views

Singularity of singularities and second microlocalization: a question that come from the stabilization of damped wave equation

In the paper [2], the Authors introduce a tool called second microlocalization, which is difficult for me. Although I have searched a lot of papers on the internet, nevertheless the material that I ...
monotone operator's user avatar
15 votes
2 answers
763 views

Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?

Fix a compact Riemannian manifold $M$ (leaving the metric implicit). What I'd like to know is if the corresponding Hodge decomposition of smooth $n$-forms $$ \Omega^n(M) \simeq \mathcal{H}^n(M)\oplus ...
David Roberts's user avatar
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2 votes
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When is an unbounded averaging operator on $\mathbb{R}\to \mathbb{R}$ closed?

Let $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{R}$. Consider the following linear operator $A$ on functions $f:\mathbb{R}\to \mathbb{R}$: $$(Af)(x) = \sum_{n=1}^\infty a_n f(x+n)+ \sum_{n=1}^\infty a_n f(...
H A Helfgott's user avatar
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3 votes
1 answer
192 views

Density beyond Stone–Weierstrass

$\DeclareMathOperator\tr{tr}$I need density assertions for spaces of polynomials which are not (that I know of) algebras. One goes like this: Fix $n\in\mathbb N$ let $S$ denote the set of self-adjoint ...
Nandor's user avatar
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41 views

Projection onto Shift Invariant Subspaces of $H^2$

Every shift invariant subspace of the Hardy space $H^2(\mathbb{D})$ is either $\{0\}$ or is of the form $\varphi H^2$ for some inner function $\varphi$. I know that if $\varphi(0) \neq 0$, then the ...
bm3253's user avatar
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3 votes
1 answer
160 views

1-1 map on the $\{0,1\}^k$

Let integer $k>0$ and let $\{0,1\}^k$ denote the set of all $1\times k$-dim vectors whose every coordinate is eithor 0 or 1, for example, $(0,1,1,0,\dots,1,0,0,1)$. For any such vector $\alpha$, ...
bc a's user avatar
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2 votes
0 answers
73 views

Extension of a tangent vector field

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
MathLearner's user avatar
1 vote
1 answer
100 views

How to show such result for generalized $ O(|x|^{-1/2}) $ function?

Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
Luis Yanka Annalisc's user avatar
6 votes
1 answer
148 views

Question on ODE involving mollifiers from Taylor's book on PDEs

In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form $$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$ with some initial condition $u(...
B.Hueber's user avatar
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28 votes
1 answer
2k views

Topology on space of hyperfunctions

This is a reference request, coming from someone with little knowledge of hyperfunctions: Which methods have been used to endow the space of hyperfunctions $\mathcal B(\mathbb R)$ with something like ...
Peter Scholze's user avatar
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1 answer
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Counterexample wanted: Banach space but not BK-space

What is an example of a Banach space that is not a BK-space? A normed sequence space $X$ (with projections $p_n$) is a BK Space if $X$ is Banach space and for all natural numbers $n$, $p_n(\bar{x}) = ...
Maulana's user avatar
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0 answers
25 views

Show that functional $J$ satisfies Palais-Smale condition iff every $P-S$ sequence is bounded

Let $H$ be a Hilbert space and let $K: H \rightarrow \mathbb{R}$ be $C^1$ and such that $\nabla K: H \rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H \rightarrow \...
YuerWu's user avatar
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2 votes
1 answer
115 views

Is there a scalar product which makes orthonormal the family of complex functions $ (f_n)_{ n \geq 1 } $?

Let $ (f_n)_{ n \geq 1 } $ be a family of complex functions defined as follow, $ \forall n \geq 1 $, $$ f_n (z) = \dfrac{1}{n^{z}} $$ I would like to ask you if it is possible to construct a ( non-...
Angel65's user avatar
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0 answers
34 views

Sufficient condition for interpolation

If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying: $Z_0=X$, $...
mejopa's user avatar
  • 101
6 votes
1 answer
220 views

Poisson kernel for the orthogonal groups

For the complex ball $|z|^2\le 1$ in $\mathbb{C}^n$, there is a Poisson kernel proportional to $|x-z|^{-2n}$. This is generalized to the unitary group $U(N)$ so that in the complex matrix ball $Z^\...
thedude's user avatar
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2 votes
1 answer
139 views

Tempered distributions at non-coinciding points and density of Schwartz functions

In the previous question, I find that situation is much less favorable than expected…. So I add more details to focus on the specific case I have in mind. Let us consider the Schwartz space $\mathcal{...
Isaac's user avatar
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0 votes
1 answer
73 views

Orthogonal space of polynomials

Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and $$ \int_0^{+\infty} f(t) t^n ...
henrysupercool's user avatar
2 votes
0 answers
48 views

References for a class of Banach space-valued Gaussian processes

Let $E$ be a separable Banach space, consider a centered $E$-valued Gaussian process $\{x_t,t\ge 0\}$ that satisfies \begin{equation} \mathbb{E}\phi(x_s)\psi(x_t)=R(s,t)K(\phi,\psi),\quad \phi,\psi\in ...
Jorkug's user avatar
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0 votes
0 answers
76 views

Maximal function on mixed $L^{p}$

Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is $$ \Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
User091099's user avatar
0 votes
1 answer
103 views

Clarification on the Interpretation of Fourier Coefficients in the Context of Fourier Projections

I am currently studying a paper (Section 3.4.3 of Lanthaler, Mishra, and Karniadakis - Error estimates for DeepONets: a deep learning framework in infinite dimensions) where the authors define an ...
Mohammad A's user avatar
1 vote
0 answers
55 views

embedding spaces of probability measures to function spaces

Let $X, Y$ be Banach spaces. I'm considering a bounded linear functional $g:X\to Y$ and its lift $g_\sharp: \mathcal{P}(X)\to \mathcal{P}(Y)$. I want to consider the inverse of $g_\sharp$ in some ...
optimal_transport_fan's user avatar
4 votes
2 answers
413 views

Is every closed subspace of the Schwartz space densely embedded into its dual space?

My original question is from this ME post but I think I need a broader understanding for this. The Schwartz space $\mathcal{S}$ and its subspaces are examples of nuclear spaces. In fact, any closed ...
Isaac's user avatar
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0 votes
1 answer
61 views

For any $p, q \in [1,\infty]$ and $s \in (0,\infty)$, can we find some $f \in L^q - W^{s,p}$?

Sobolev inequalities show us when we can embed a Sobolev space into another. However, I wonder if these inclusions are always proper. More specifically, let $\Omega \subset \mathbb{R}^n$ be a bounded ...
Isaac's user avatar
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4 votes
1 answer
119 views

Multivariate polynomial approximation of functions in Sobolev space

I found a result of the estimation error of polynomial approximation in page 6 of https://scg.ece.ucsb.edu/publications/theses/ARajagopal_2019_Thesis.pdf The statement is for $f \in W^{k, p}\left([-1,...
Iris's user avatar
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3 votes
0 answers
55 views

Rate of convergence of mollified distributions in Besov spaces with negative regularity

Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
Marco's user avatar
  • 293
4 votes
0 answers
99 views

Larger possible chain of closed subspaces in the dual of a Banach space

In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces. My question is the following. If $X$ is an ...
Emerick's user avatar
  • 53
1 vote
1 answer
122 views

Form of a hereditary subalgebra of $C^*$-algebra $C_0(X)$

I would like to show that: "every hereditary subalgebra $U$ of a $C^*$-algebra $C_0(X)$ for a locally compact Hausdorff Space $X$ has the form $J_E := \{f \in C_0(X) : f|_E=0 \}$ for a closed ...
VvvV's user avatar
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3 votes
0 answers
76 views

Excising the trace of a $II_1$-factor

Recall that a state $\varphi$ on a $C^*$-algebra $A$ is said to be excised by projections if there exists a net of projections $e_i \in A$ such that $\| e_i a e_i - \varphi(a) e_i\| \to_{i} 0$ for all ...
pitariver's user avatar
  • 297
8 votes
1 answer
243 views

Characterizing germs of smooth functions

There's a sheaf of smooth real-valued functions on $\mathbb{R}$, and its germ at $0$ is some vector space $V$. I would like to understand this space. There is a surjective linear map $$ \phi \colon ...
John Baez's user avatar
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2 votes
1 answer
132 views

De la Vallée Poussin criterion on uniform integrability for infinite measures

The de la Vallée Poussin criterion (which is often used in combination with the Dunford-Pettis theorem) is usually formulated for probability measures/finite measures, for example in [Bogachev: ...
kumquat's user avatar
  • 85
0 votes
0 answers
85 views

Dual of closure

Currently I'm studying about abstract interpolation theory for my research. One of the basic ways to construct new interpolation spaces, given an interpolation space $E$ with respect to a compatible ...
Guillermo García Sáez's user avatar
-1 votes
0 answers
33 views

Relationship between Hankel function and 2D Helmholtz equation

To solve this the 2D Helmholtz equation $(\nabla^2 +k^2)G(\vec{r})=\delta(\vec{r})$ I use the Fourier transform and I get this integral $ G(\vec{r}) = -\int \frac{d^2q}{(2\pi)^2} e^{i\vec{q}\cdot \vec{...
leopoldo villoni's user avatar
-1 votes
0 answers
87 views

Sufficient conditions for $1/f\in L^p$

This is a very simple question that I have not found a satisfactory answer for. When is the reciprocal of a function $f:[-1,1]\to\mathbb{R}$ in $L^p([-1,1])$? In other words, when is $\int1/f^p\,dx<...
tedl445's user avatar
0 votes
0 answers
49 views

Compiling a List of Schauder Bases for Function Spaces

Schauder bases are fundamental tools in functional analysis, allowing us to represent functions within a space as infinite sums of basis elements. These bases play a crucial role in studying the ...
ALIN's user avatar
  • 21
1 vote
1 answer
213 views

When is this operator positive semi-definite?

I have the following operator $$\Phi(\chi_A)=\int \text{d}\eta\, \text{d}\zeta\,\chi_A(\eta,\zeta)\,e^{i(\eta \hat{P}+\zeta\hat{Q})}.$$ With $\chi_A$ the indicator function associated to a set $A\...
Nicolas Medina Sanchez's user avatar
0 votes
1 answer
197 views

A variation of the Riesz Lemma

Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...
Emerick's user avatar
  • 53
0 votes
0 answers
57 views

The size of super level sets and the symmetry on a sphere

Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define $$ S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}. $$ Suppose ...
MathLearner's user avatar

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