Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,437 questions with no upvoted or accepted answers
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Are invertible measures strictly dense?
Let $L_1(\mathbb T)$ be considered as a closed ideal of $M(\mathbb T)$, the Banach algebra of measures on the circle. Then $M(\mathbb T)$ can be identified with the multiplier algebra of $L_1(\mathbb ...
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Density of squares of radial eigenfunctions
The eigenfunctions of the Laplace operator on the disc can be written in polar coordinates as $f(r,\theta)=R_{nk}(r)e^{ik\theta}$, where $k\in\mathbb Z$ and $n\in\mathbb N$ and the radial function is $...
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Spectral properties of Non-local Differential operators on real line
I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs.
Definition: A ...
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Interchange of supremum and integral
Let $f : X \to Y$, $X \subset R^n$, $Y$ Banach space, $g : X \times Y \to R \cup \{ \infty \}$, $L^n$ the n-dimensional Lebesgue measure.
Are there some results under which the following interchange ...
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Unitary representations of finite dimensional Lie groups on infinite-dimensional Hilbert spaces
I am interested in the proofs of continuity of some standard unitary representations appearing in Physics. Additionally, I am interested in the integration of finite-dimensional Lie algebras of skew-...
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Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
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Spectrum of perturbed differential operators
I am looking for a reference that could help me with the following two questions:
Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary. Consider a sequence of differential ...
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Covariance operator analogue for manifolds and respective measure manifolds
Assume $E$ is a connected riemannian manifold with geodesic metric space structure given by $d$ and $P$ is a probability measure over $E$ with Borel sigma-algebra given by this metric structure. Also ...
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798
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What is the Banach dual of the Bochner space $L^\infty(\Omega;X)$?
Suppose $\Omega$ is a $\sigma$-finite measure space (I'm happy to take $\Omega = \mathbb{N}$) and let $X$ be a Banach space. It's pretty well known that the Banach dual of $L^\infty(\Omega)$ can be ...
user14166
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Generalized singular numbers and the Haagerup $L^p$ spaces
Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...
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Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term
Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...
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Extension operators for topological vector space-valued smooth functions on closed sets
There are many known results about extension theorems for real-valued functions on closed sets, with varying levels of differentiability and so on, all very roughly following the Whitney approach. For ...
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Constructing Extreme Points in Reflexive Banach Spaces
A theorem of Lindenstrauss and Phelps states that if $X$ is a separable reflexive Banach space then the unit ball of $X$, $Ba(X)$, has uncountably many extreme points. The proof goes by contradiction ...
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Chord-arc property of n-tuples of commuting operators
Assume that we have an $n$-tuple $S^0=(S^0_1,\dots,S^0_n)$ of commuting operators in a Hilbert space $H$ and another such an $n$-tuple $S^1$. Is it possible to connect these two $n$-tuples by a ...
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Interpolation between $H^1$ and $H^1\cap L^1$
Suppose that $T:H^1(\mathbb{R}^3)\rightarrow\mathbb{R}$ is a linear bounded operator, with operator norm $M_2$. In particular, given $1\leq p\leq2$, there exist optimal constants $M_p\leq M_2$ such ...
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Integral-like concepts
I am looking for interesting concepts (I guess you could say functionals from the function space $[a;b]\to\mathbb R$) that are like integrals in some respect.
The background is that I have proven a ...
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Categorical description of dense homomorphisms of topological algebras
Let $A$ and $B$ be topological associative algebras (no matter, in which sense, for example, over $\mathbb C$, with identity, and with separately continuous multiplication). Let us say that a (...
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Basis functions for approximation of a convex function on unit simplex
Consider the unit $D$-simplex $S^D=\left\lbrace (x_0, x_1, \ldots, x_D) \in \mathbb{R}^{D+1} \mid \sum\limits_{i=0}^{D}x_i = 1, x_i \geq 0 \right\rbrace$. I have a bounded, convex function $f:S^D\to\...
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Is there a decomposition strengthening of the Sauer-Shelah Lemma?
Let $S \subset \{-1,1\}^n$. For a subset $A \subset [n]$ let $P_A$ denote the coordinate projection operator on S; in other words let $P_A(S)$ be the coordinate projection of $S$ onto the coordinates ...
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Reference for the Banach Manifold structure of $C^k(M,N)$
I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:
Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^...
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Real interpolation space between the Wiener algebra and $L^2$
The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for $\...
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Max min of functionals
I have an interesting question which I believe was probably already studied, but I could not find anything. Let $n, m \geq 1$ be fixed. Suppose that $|| \cdot ||$ is a norm in $\mathbb{R}^n$ and $f_1, ...
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Proof that the second Borel cohomology group of $(\mathbb R, +)$ is trivial
Does anyone have a reference for a fairly direct proof that the second Borel cohomology group for $(\mathbb R, +)$ (with the trivial action on the circle group) is trivial? The motivation is to show ...
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Operator arithmetic-harmonic mean inequality with operator-valued weights
Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...
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Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space
Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space.
We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...
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243
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A compactification of the non-negative rationals with the discrete topology
Let $S$ be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let $B=\ell_\infty(S)$; as a ${\rm C}^*$-algebra this is ...
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a question about Tsirelson's space
NOTE: I asked this question over at math.stackexchange.com but got no answer or comments after 3 days, probably because it's a bit specialized. Hopefully it is interesting enough to ask over here.
...
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Which functions can be approximated by piecewise constant functions?
Let $\Omega \subset \mathbb{R}^d$ be a connected and bounded domain. We call a function $f\in L^\infty(\Omega)$ nice if for each $\epsilon>0$ there exist $n\in \mathbb{N}, a_1,\dots,a_n \in \mathbb{...
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Products of spaces containing no copies of $\ell_2(\Gamma)$
Given an infinite set $\Gamma$, I would like to know if the class of Banach spaces containing no copies of $\ell_2(\Gamma)$ is stable under finite products.
When $\Gamma$ is countable the answer is ...
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Ultrapowers and complemented subspaces
Let $Y$ be a closed subspace of a Banach space $X$, and let $\mathcal{U}$ be a nontrivial ultrafilter on the set $\mathbb{N}$ of all integer numbers.
It is not difficult to see that if $Y$ is ...
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Pettis Integrability and Laws of Large Numbers
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
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Explicit form of the homeomorphism between $C[0,1]$ and $C[0,1]\setminus 0$
How to construct the homeomorphism between $C[0,1]$ and $C[0,1]\setminus\{\theta\}$
in the explicit form?
Here, as usual, $C[0,1]$ is the Banach space of continuous functions $f:[0,1]\to\mathbb{R}$ ...
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Norms on tensor products
Let $A$ be an algebra of operators on a Hilbert space $H$. Let $A^m$ and $A^n$ be free $A$-modules with bases $e_1, \ldots, e_m$ and $f_1, \ldots, f_n$. Define a norm on $A^m \otimes A^n$ as follows:
$...
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Given that a conditional measure is Gaussian, how bad can the original measure be?
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
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Paving conjecture for Toeplitz matrices
Let me first recall what is the so-called paving conjecture:
for any $\epsilon >0$, there exists $r\in \mathbb N$ such that
for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a ...
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Terminology for notion dual to "support"
If $X$ is a set (feel free to think of it as finite, but it doesn't have to be) and $f$ a real function on $X$, call the support $\operatorname{supp} f$ the subset of $X$ consisting of all elements $i\...
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Ricci-flat metrics on Cotangent bundles in adapted complex structure
greetings,
Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...
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Birth-Death Process associated with Orthogonal Polynomials
I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...
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distributions on Lie groups and representations
Let $G$ be a Lie group and $\pi$ a continuous action on $V$, a Fréchet space.
This action induces a representation of the space of compactly supported functions, $C_c(G)$, with convolution as product ...
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What is the intersection of the closures of left invertible operators and right invertible operators?
From Douglas Zare's answer (see Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?), one know that
$$ \overline{G_{l}(X,Y)} \bigcap \overline{G_{r}(X,Y) } = \...
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Do the translates of integrable function approximate its radial part?
For an integrable function $f$ on $\mathbb R^n$ we consider its ``radial'' part
$$R(f)(x)=\int_{\mathrm{SO}(n)} f(kx)dk.$$ What is the minimal condition on $f$ so that the span of translates of $f$ (...
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What is the structure of a space of $\sigma$-algebras?
Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
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Local minimum from directional derivatives in the space of convex bodies
I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...
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Dual space of continuous functions
Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
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Square and cube?
Gowers and Maurey proved in their remarkable paper(s), that there is a Banach space $X$ such that $X$ is isomorphic to its cube $X\oplus X\oplus X$ but not to isomorphic to its square $X\oplus X$. ...
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Projective and injective tensor product
It is well known that for arbitrary Banach spaces $X$ and $Y$ we have that the dual space
$(X \hat{\otimes}_{\pi} Y)^* = \mathcal{L}(X, Y^*)$.
If we take $\ell^p$ and $\ell^q$ such that $p < q^{\...
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$f(x) \ne g(x)$ but $f(f(x))=g(g(x))$ - is there a name/some discussion of this property?
In the context of iteration of functions I look at the eigenvalues of the associated matrixoperator/Carleman-matrix .
If a function $\small f(x)$ has a negative eigenvalue in its associated ...
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Is there an idempotent measure on the free LD system?
This is a follow up question to MO question "Idempotent measures on the free binary system?".
Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law:
...
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Finite element method inverse estimate
$\DeclareMathOperator\diam{diam}$Looking for a proof in the literature of the following lemma:
Let $K\subset\mathbb{R}^d$ be a bounded domain. Let $P_X$ be a finite dimensional subspace of $\mathcal{...
- 3,217
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Spectrum of an operator arising in a dynamical problem
(Question edited according to Denis Serre comment).
While studying the action of dilating map of the circle on probability measures, I ran across the following operator:
$$\mathcal{K}^* : L^2_0(\mu)\...
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