Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,436 questions with no upvoted or accepted answers
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Arithmetic progressions and removal lemmas for graphs in arithmetic combinatorics
As it is well known, one can gets a proof of Roth's Theorem concerning arithmetic progressions of length 3 (APs for short) by using the celebrated Ruzsa-Szemerédi triangle removal lemma for graphs.
In ...
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212
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Two equivalent definitions of semisimplicity of group representations, proof by Zorn's lemma, a “counterexample” from the Fourier transform theory
Consider a representation $A$ of a group $G$ in a complex vector space ${\mathbb{V}}$:
$$
A:~~G~\longrightarrow~\operatorname{GL}({\mathbb{V}})~~,
$$
and let ${\mathbb{V}}$ be decomposable into a ...
- 603
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99
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Definition clarification: "regular directed distributions"
(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of ...
- 1,171
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449
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Equivalence of weak and weak-* convergence for sequences and reflexivity
Let $X$ be a Banach space and $X^*$ its topological dual space. Let us define the property (WS):
For all sequences $(x_n^*) \subset X^*$ and all $x^* \in X^*$, we have
$$x_n^* \rightharpoonup x^* \...
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126
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A path with zero increments and positive area
I am studying rough paths from the 2007 St Flour lecture notes and I came across the example at the end of chapter one of the sequence of paths $X(n):[0,2\pi]\to \mathbb R^2$ given by $X_t(n) = \frac{...
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205
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Beurling's theorem on invariant subspaces
Beurling's theorem characterize the closed subspaces $M\subset H^2$ of the Hardy space, which are invariant under the shift operator $Sf(z):=zf(z)$, as spaces of the form $\varphi H^2 $ where $\varphi$...
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154
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Is it possible to reconstruct the universally measurable sets in X from the $C^*$-algebra $C(X)^{**}$?
This continues my question of two months ago. Let $X$ be a compact Hausdorff topological space. We consider the $C^*$-algebra $C(X)$ of continuous functions on $X$, its dual space $C(X)^{*}=M(X)$ of ...
- 7,426
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81
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Example of the bounded convolution operator when Sharpley's conditions does not hold
I am reading about Orlicz and Marcinkevich spaces, and wondering whether there is an example in which Sharpley's condition is not satisfied for a special bounded operator $T_k$ (see for reference ...
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135
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Holmgren's theorem on the boundary
Consider $\Omega$ a bounded Lipschitz domain, with $\gamma \subseteq \partial \Omega$ a $C^2$ manifold. I am interested in proving the following.
Let $u: \Omega\times [0,T]\rightarrow \mathbb{R}$ be ...
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130
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Is the range of the exterior covariant derivative closed in $L^{2}$?
Let $(M,g)$ be a compact Riemannian manifold. Given a tensor bundle $\mathbb{E}$, let $\nabla:\Gamma(\mathbb{E}) \rightarrow \Gamma(T^{*}M\otimes \mathbb{E})$ be the canonical connection induced by ...
- 808
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174
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On continuous seminorms on Fréchet-Stein algebras
Let $K$ be a discretely valued complete non-archimedean field and $U$ be a left Fréchet-Stein algebra as defined in Algebras of p-adic distributions and admissible representations, with a Fréchet-...
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220
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Can any POVM be induced by a quantum instrument?
I suspect this is the obvious result of something in operator algebras, but that's far outside my field.
Recall that a projection-valued measure is a map $E$ from a sigma-algebra $\mathcal{F}$ on ...
- 854
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110
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Tauberian theorem with flatness condition
Suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is a series with $a_n\in \mathbb{R}$ and radius of convergence $1$ and such that $f$ restricted to $[0,1[$ admits a smooth extension to $[0,1]$ with $f^{(n)}(1)...
- 278
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257
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Complex Hölder space
I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
- 136
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430
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What's the problem with the evaluation map not being continuous?
When introducing differentiable functions between locally convex spaces, many authors (e.g. Bastiani, Keller, Kriegl-Michor) notice that the evaluation map
$$ E\times E^*\to\mathbb R,\qquad (x,L)\...
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190
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$C^1$-regularity of solution of a Dirichlet problem
I've been stuck trying to hunt down a proof/reference for a certain regularity result which I need for my thesis (all the authors I've consulted refer to it as "well-known"). Consider the ...
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168
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Besov space norms
We need to recall some Besov space norms to formulate the question.
Let $d \in \mathbb N$, $0<s<2, 1 \le p,q \le \infty$. Then the Besov space $B^s_{p,q}(\mathbb R^d)$ is given by the norm
$$ \...
- 475
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108
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On the derivatives of the solutions of the heat equations with Neumann boundary condition
Let $\Omega$ be a smooth domain with compact closure. More precisely, $\Omega$ is a $C^\infty$ domain. We write $\partial \Omega$ for the boundary of $\Omega$, and denote by $n$ the inward unit ...
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465
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Orthonormal basis of eigenvectors of Hamiltonian - Is there any theorem justifying the physicist approach?
In his book The Principles of Quantum Mechanics, Dirac states:
"We call a real dynamical variable whose eigenstates form a complete set an observable."
To Dirac, any observable has a ...
- 1,305
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278
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Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure
Let $(X, | \cdot |)$ be a Banach space.
I am interested in whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
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73
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Does the incompressible Navier-Stokes equation have a smooth solution if the initial vorticity is smooth and $p$ integrable for any $p$?
Consider the incompressible Navier-Stokes equation on $(0,T)\times \mathbb{R}^3$ for fixed $T>0$.
If the sequence of mollified initial vorticities $(\omega_0^{\nu})_{\nu}$ is uniformly bounded in $...
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233
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Is it possible to reconstruct the compact space $X$ from the space of measures $M(X)$?
Let $X$ be a compact Hausdorff topological space and $C(X)$ the Banach algebra of continuous functions $u:X\to\mathbb C$ (with the usual $\sup$-norm). It is well-known that the structure of Banach ...
- 7,426
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55
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Injectivity of a convex combination of squares $A^*A$ in $\ell^\infty$
Consider two operators $A,B: \ell^p \to \ell^p$ ([defined and]bounded for all $p \in [1,\infty]$) as well as their adjoints $A^*,B^*:\ell^p \to \ell^p$. Assume $A^*A$ and $B^*B$ have trivial kernel ...
- 4,432
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209
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Dominated convergence theorem for Banach limits
The notion of a Banach limit is usually defined for the space of bounded sequences, but one can define it for more general spaces (see "What is a generalized limit?" and "Do ...
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77
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Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$
My research needs help in finding examples of unitary matrices $U$ which have entries
\begin{align}
U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
- 41
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68
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Relationship between Hardy-Orlicz space and the corresponding Orlicz space
For $p \in [1, \infty]$ the Hardy space $H_p$ is defined as the space of all analytic functions $f$ on the open disk satisfying
$$\|f\|_{H_p} = \sup_{0 < r < 1} \|f(r\cdot)\|_{L_p(\mathbb{T})} &...
- 565
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139
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Continuity of parameter integral
Given two compact Hausdorff spaces $K$ and $L$, a bounded and separately continuous function $f:K\times L\to \mathbb C$, and a complex measure with finite variation $\mu$ on $L$ endowed with the Borel ...
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A weighted $W^{2,p}$ estimates
Let $\Omega$ be a bounded smooth domain and $u\in W^{2,p}(\Omega)\cap H^1_0(\Omega)$. By the classical $L^p$ theory of second order elliptic equation, we have
$$
\|\nabla^2u\|_{L^p(\Omega)}\leq C\|\...
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Asymptotic uniform convexity conditions for subsets of the $B_X$
The following question is relatively straightforward and almost looks like an exercise from a textbook but I have no idea how to handle it. The problem is related to spaces with asymptotically ...
- 1,073
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324
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Would you help me to find this expression?
I'm studying a paper, which I'll include a small piece here. And I'm struggling to calculate
$$C_n\|u_{m,n}\|^{\left(\frac{2*}{2}\right)^k\frac{2*-q}{(r_k)^k}}_{L^{2*}(\Omega)}$$
Where $\Omega$ is an ...
- 63
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Dual space of Carleman functions
Let $X$ be the space of all weakly measurable functions $\gamma:\mathbb{R}^n \to L^2(\mathbb{R}^n)$ (modulo functions that are 0 almost everywhere) for which
$$\|\gamma\|_X^2 := \sup_{\|g\|_{L^2}=1} \...
- 141
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108
views
$ f,g\in \mathrm{VMO} $ but $ f\cdot g\notin \mathrm{VMO} $
We say a function $ f\in L^1_{\mathrm{loc}}(\mathbb{R}) $ is in $\mathrm{BMO}(\mathbb{R})$ if
$$\|f\|_{\mathrm{BMO}}=\sup_{I}\frac{1}{|I|}\int\limits_I |f(y)-f_I|\, dy<\infty$$ for all intervals $I\...
- 1,219
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159
views
$L^\infty-L^\infty$ bounds for heat semigroups constructed from the Dirichlet Laplacian
Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding ...
- 1,407
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171
views
Covering number $C^k$-balls in $C(\mathbb{R}^n)$
Fix a positive integer $n$ and and an non-negative integer $k$. The Arzela-Ascoli theorem guarantees that for a given positive integer $k$ and a given $L>0$ the set
$$
Ball_{C^{k,1}([0,1]^n)}(0,L)
...
- 5,405
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171
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A version of the Nash-Moser inverse function for unbounded domains?
Do there exist versions of the Nash-Moser inverse function theorem applicable to spaces of unbounded smooth functions on unbounded domains in $\mathbb{R}^n$? Any reference would be appreciated but ...
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Extended adjoint of Volterra operator
Let $V$ be a Volterra operator on $L^2 [0,1]$.
Does there exist a nonzero operator $X $ satisfying the following system
$VX=XV^∗$, where $V^∗$ is the adjoint of the Volterra operator?
$$ V(f) (x) =\...
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252
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Characterization of a Bochner/strongly measurable function solely as a random element
Be $(\Omega, \mathcal{A}, P)$ and $E$ a probability space and a Banach space respectively.
This paper of G.A. Edgar contains a proof that, for a function $X: \Omega \rightarrow E$, being weakly ...
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Unital commutative dual Banach *-algebras whose $w^*$-closed ideals are principal
Let $A$ be a commutative Banach *-algebra. For a given ideal $I$ of $A$, we say that, it is principal if there is a projection $p$ (i.e. $p^2=p=p^*$) in $A$ with $I=Ap$.
Q. Any characterization ...
- 4,058
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276
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Schur-Horn theorem for principal submatrices
The Schur-Horn theorem says that there exists a Hermitian matrix with diagonal entries $d_1,d_2,\ldots,d_n$ and eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_n$ if and only if $(\lambda_1,\lambda_2,\...
- 6,351
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If $u_n\rightharpoonup u$ in $L^2(0,T;L^2)$ then there is a subsequence such that $u_n(t)\rightharpoonup u(t)$ almost everywhere?
If $u_n\rightharpoonup u$ in $L^2(0,T;L^2(\Omega))$. Can we find a subsequence such that $u_{n_k}(t)\rightharpoonup u(t)$ almost everywhere on $[0,T]$?
I'm not sure if this question is trivial or not,...
- 153
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Non-emptiness of spectrum $\sigma(a)$ in non-Archimedean Banach algebras
I'm trying to determine whether or not the standard proof that the spectrum of a point in a unital Banach algebra is non-empty can be adapted to prove the same thing over certain non-Archimedean ...
3
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229
views
$f(x)>0$ and $f(y)>0$ implies $f(x+y)>0$, then there must exist an linear function $g$ such that $g(x)>0$ iff $f(x)>0$?
Background: Let $x,y\in\mathbb (0,+\infty)^n$. $f$ is a continuous function on $\mathbb R^n_+=(0,+\infty)^n$. Consider the following condition (1), the sign of $f(x+y)$ is dependent on the sign of $f(...
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Stinespring's theorem: can we choose the dilation to be an isometry?
Let $A$ be a $C^*$-algebra and $\varphi: A \to B(H)$ be a completely positive contractive map. Stinespring's theorem says that there exists a $*$-representation $\pi: A \to B(H')$ and a bounded ...
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187
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Analogue of Kolmogorov/Arnold superposition for general manifolds?
Previously asked and bountied at MSE with slightly different language:
Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
- 20.7k
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282
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Left ideals of $\ell^{\infty}(A)$ containing all weakly null sequences in a Banach algebra $A$
Let $A$ be a Banach algebra. $\ell^{\infty}(A)$, the space of all bounded sequences in $A$, is a Banach algebra with pointwise operations.
Let $w_0(A)$ be the subspace of all weakly null sequences in $...
- 2,605
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117
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Multiplier on a Sobolev space
Let $b$ be a function and $W^{1,2}$ the first-order Sobolev space on some Euclidean space (edit: so the whole space, but in arbitrary dimension, although $d=1$ would be interesting for me for the ...
3
votes
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80
views
Every Borel linearly independent set has Borel linear hull (reference?)
I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone.
Theorem. The linear hull of any linearly independent Borel set in a Polish ...
- 42k
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120
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Approximation of a linear functional by linear continuous functionals
Let $X$ be a locally convex space, $T$ a balanced convex compact set in $X$, and $f:X\to\mathbb{C}$ a linear functional which is (not necessarily continuous on $X$, but) continuous on $T$. It is not ...
- 7,426
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votes
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answers
323
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Does convolution by a Schwartz function preserve symbol classes?
I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
- 395
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154
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What do people call functionals on holomorphic functions and on polynomials?
There are four most important functional spaces in analysis:
the space $\mathcal{C}(M)$ of continuous functions on a topological space,
the space $\mathcal{E}(M)$ of smooth functions on a smooth ...
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