Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,246
questions with no upvoted or accepted answers
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Isometries on the unit sphere
Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?
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201
views
Understanding the odd-dimensional index
Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...
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346
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What is known about "almost orthogonal vectors"?
Motivation:
Suppose we have a kernel $k(a,b)$ defined over the natural numbers.
Then by the Moore–Aronszajn theorem, we can embedd the natural number $a$ in some Hilbert space $\mathbb{H}$, which we ...
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84
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Analogue of Friedrichs extension for Hilbert $C^*$-modules
Suppose one has a densely defined symmetric operator $T:\mathcal{M}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a Hilbert $A$-module for a $C^*$-algebra $A$. Suppose that $T$ is non-negative, so ...
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Does this ideal in $B(L_1)$ have a (bounded) right approximate identity?
I will take a roundabout way to defining this ideal, because (a) this route is how my collaborators and I came to it (b) this alternative definition, rather than the standard one, may suggest a ...
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411
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(geodesic) smoothness of f-divergence with respect to the Wasserstein metric
We consider the f-divergence, which takes the form
$$
D_f(P \| Q) = \int_\Omega f\left(\frac{dP}{dQ}\right) dQ.
$$
For example, when $f(t) = t \log t$, we obtain the KL-divergence.
My question is ...
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177
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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-...
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311
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Status of two Banach space theory open problems posted by Pełczyński
In the book 'Open Problems in the Geometry and Analysis of Banach Spaces', I am interested in the following two problems.
Problem $1$: Let $X$ be a separable infinite-dimensional Banach space that is ...
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200
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Equivalent strictly convex norms in spaces of small density
Can one construct in ZFC a Banach space of density character $\omega_1$ that does not have an equivalent strictly convex norm?
Maybe one may apply some kind of a Löwenheim–Skolem-type argument to a ...
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221
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Loomis-Whitney versus Gagliardo inequalities
When searching for a reference, I discovered a curious fact about the Wikipedia page concerning the Loomis-Whitney Inequality (LWI).This page, which exists only in an English version, states that the ...
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250
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Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$
I have found the following claim made very clearly at least once in the published literature (see below):
Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...
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568
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Lavrentiev Phenomenon
Does there exist a (onedimensional) integral functional of calculus of variations
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that not only
$$
\inf_{y\in Lip([a,b])}F(y)>\inf_{y\in W^{1,1}([a,b])...
7
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502
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Counter-example to the completeness of the Wasserstein metric
$\newcommand{\P}{\mathcal{P}}$
Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
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461
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Characterizing the sum $L^1 + L^\infty + L^{1,\infty} + L^{\infty, 1}$ of iterated Lebesgue spaces "by duality"
For the usual Lebesgue spaces $L^p (\mu)$ ($p \in [1,\infty]$) on a ($\sigma$-finite) measure space $(X,\mu)$, it is well-known that one has the characterization
$$
L^p (\mu) = \left\{f : X \to \Bbb{...
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1k
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Books on von Neumann algebras
I am interested in non-commutative $L^p$ spaces. I have a very basic background on von Neumann algebras. But all the papers appearing now a days really requires very deep knowledge of von Neumann ...
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130
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Different definitions of fractional sobolev spaces
Let $\Omega$ be a bounded and smooth domain in $\mathbb R^d$. For any $s\in (0,1)$ we can define $H_s(\Omega)$ to be the space of functions $u\in L^2(\Omega)$ such that $$(x,y)\mapsto \frac{|u(x)-u(y)|...
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What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
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181
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distance distributions on a hypersphere?
Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let
$\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define
$$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$
where ...
7
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299
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Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories
Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
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243
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Commutation preserving operators
Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve ...
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178
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Is there a quotient of $c_0$ without the approximation property?
The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property?
This would follow if ...
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133
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Kolmogorov superposition representation of Lipschitz functions
Kolmogorov's superposition theorem states that
if $f:[0,1]^n \to \mathbb{R}$ is an arbitrary continuous function, then it has the representation
\begin{align}
f(x) = \sum_{q=0}^{2n} \Phi_q (\sum_{p=1}...
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241
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A "slice-map" type problem for symmetric tensors in the square of a nuclear C*-algebra
Throughout: let $\otimes$ denote the minimal (i.e. spatial) $\newcommand{\Cst}{{\rm C}^*}\Cst$-tensor product of two $\Cst$-algebras.
Let $B$ be a unital, nuclear $\Cst$-algebra and let $A\subset B$ ...
7
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267
views
Approximation in the tensor square of a weakly exact von Neumann algebra
Background. I think I can prove something about a certain construction definition for Fourier algebras of discrete groups, under the assumption that the group is exact (well, really I use Yu's ...
7
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answers
862
views
What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?
I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$ \operatorname{supp} \phi \...
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727
views
What function space does holomorphic functional calculus give us?
Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function $f:U\rightarrow\...
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551
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The Banach space of bounded functions with countable support
Let $X$ be a set of cardinality $\aleph_1$ and consider the Banach space $\ell_\infty^c(X)$ of all scalar-valued bounded functions on $X$ which are non-zero only for countably many elements of $X$ ...
7
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116
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Bundles over Function Spaces
Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable Hilbert-...
7
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286
views
Trace class norms of special integral operators
Let $\mu$ be a finite compactly supported Borel measure on the real line. On the space $L^2(\mu)$ consider the integral operators
$$
(K_a f)(x)=\int k_a(x, y)f(y)d\mu(y)
$$
with
$$
k_a(x, y)=\frac{a\...
7
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233
views
Is there a tensor norm that preserves Rosenthal Banach spaces?
By a Rosenthal Banach space I mean one that does not contain an isomorphic copy of $\ell_1$. Is there a tensor norm $\alpha$ such that the Banach tensor product $E\otimes_\alpha F$ is Rosenthal if $E$ ...
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283
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Complex interpolation of a Banach space and its antidual when the space has a basis
Given a complex Banach space $X$ and its antidual $\hat X^*$, it is possible in some cases to apply the complex interpolation method, and get as $(X,\hat X^*)_{1/2}$ a Hilbert space. See [F. Watbled, ...
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1k
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Prokhorov's theorem for finite signed measures?
Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure.
Notation used ...
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191
views
Central Extension of Continuous Loop Group
For the group $LG$ of smooth loops into a simple compact 1-connected Lie group $G$ there is a well-known universal central extension. My qustion is basically whether this extension also exists for the ...
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689
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Minkowski's Inequality for Integrals in Orlicz spaces
EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.
Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0,\infty)\to[0,\infty)$...
7
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284
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Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
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342
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An $L^{\infty}$ version of principal component analysis?
I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal.
I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...
7
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264
views
Problem with Shelah and Stern's paper on the Hanf number of the theory of Banach spaces
I have been trying to understand "The Hanf number of the first order theory of Banach spaces" by Shelah and Stern (Trans. AMS 244 (1978) 147-241). They construct a normed space $M$ from a Hilbert ...
7
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612
views
"Liftings" of L^\infty functions
This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...
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725
views
The log kernel and Bochner Theorem
I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that
$$
L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t)
$$
for every $x\in [0,1/2]$.
On a structural ground, this ...
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answers
137
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integral transforms defined by polygons
Are there any literature on operators of the form
\[ (Tf)(x) = \int K(x,y)f(y), dy\]
where $K(x,y)$ is the characteristic function of a polygon in $\mathbb{R}^2$. I would like to know if the spectrum ...
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156
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Seeking reference - criterion for the existence of a positive linear functional on an ordered vector space below a given function
The following surely appears somewhere, I would greatly appreciate a reference. (The aim is to get a measure via Riesz representation, but that has nothing to do with the statement.)
Let $X$ be an ...
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1k
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Reference request: Arzela-Ascoli theorem for smooth Hölder norms
Could anyone suggest a textbook account of the Arzela-Ascoli theorem for $C^{k,\alpha}$ norms?
6
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241
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Applications of Banach space homology
There is a well-developed theory of Banach space homology. What are some of its useful applications to Banach space theory and which important questions can one answer using it? In other words, how ...
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196
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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 $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
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91
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Conditions for completely positive maps to act homomorphically across multiple subalgebras
For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....
6
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211
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Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
6
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171
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Dependence of Neumann eigenvalues on the domain
I have the following problem in hands, in the context of a broader investigation:
Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following:
For any $\...
6
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227
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Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?
QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...
6
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0
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149
views
Gaussian lower heat kernel bounds on non-convex bounded domain
I am looking for a proof the following theorem.
Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
6
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0
answers
195
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Detailed examples of induction on scale
I'm trying to understand the induction on scale argument in harmonic analysis. On this abstract it's mentioned that induction on scale can be used to prove Cauchy Schwartz inequality, Beckner's tight ...