Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
873
questions
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3
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Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$
Let $X$ be a Banach space. I think that some time ago I read somewhere that, in general, the space $\ell_2(X)$ of all sequences $(x_n)$ in $X$ with $\sum_{n=1}^\infty \|x_n\|^2<\infty$ is not ...
12
votes
3
answers
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To what extent is convexity a local property?
A polyhedron is the intersection of a finite collection of halfspaces. These halfspaces are not assumed to be linear, i.e. their bounding hyperplanes are not assumed to contain the origin. The ...
12
votes
1
answer
562
views
Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...
12
votes
0
answers
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Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?
Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...
11
votes
2
answers
703
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A neat evaluation of an infinite matrix?
Let $M_n$ be an $n\times n$ matrix defined as
$$M_n
=\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$
With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. ...
11
votes
2
answers
904
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Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...
11
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1
answer
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Stone-Weierstrass analogue for $L^p$
Let $A$ be a complex algebra of bounded measurable functions on the measure space $(X,\mu)$ (case of $[0,1]$ with Lebesgue measure is enough for me) closed under conjugation. Assume that $A$ separates ...
11
votes
1
answer
237
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Bilinear product of two summable families
Consider the following statement, which I suspect is false as written:
Let $E,F,G$ be (Hausdorff) topological vector spaces (over $\mathbb{R}$), let $\varphi\colon E\times F\to G$ be continuous and ...
10
votes
2
answers
878
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Continuity of the product map
Let $A$ be a $C^*$-algebra.
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$
is continuous with ...
10
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1
answer
738
views
When do tensor products of C*-algebras commute with colimits?
Let $I$ be a filtered poset, which you should think of as being huge. Let $A_i$ be an $I$-diagram of $C^{\star}$-algebras and let $A$ be the colimit of this diagram; if necessary, we can also assume ...
10
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2
answers
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Do Hausdorff locally convex inductive limits always exist?
The following is from Schaefer, "Topological Vector Spaces", 1999, p. 56/57:
Let $(E_\alpha)_{\alpha \in A}$ be a family of locally convex spaces with $\alpha$ in a directed poset $A$ and $h_{\beta \...
10
votes
1
answer
462
views
Which W*-algebras are the duals of C*-coalgebras?
A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...
10
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4
answers
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Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces
This question was posed on MathStackExchange but did not get an answer (even with a bounty).
In all books that I have checked the spectral theorem (every self-adjoint unbounded operator on a Hilbert ...
9
votes
0
answers
717
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Reference request : Grothendieck's topological space valued integral
As I am learning the different kind of Banach space valued integrals (Pettis, Bochner), I know that Grothendieck made a "mémoire" in his youth about this topic, but I don't know if it is available ...
9
votes
1
answer
919
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A problem in functional calculus
This is embarrassing, I think it must work, but I can't see how to prove it works. If anyone knows enough functional calculus of operators on a Hilbert space to tell me how to do it, I would be very ...
8
votes
2
answers
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VC dimension, fat-shattering dimension, and other complexity measures, of a class BV functions
I wish to show that a function which is "essentially constant" (defined shortly) can't be a good classifier (machine learning). For this i need to estimate the "complexity" of such a class of ...
8
votes
2
answers
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Relating different topologies on $C^{\infty}_c(M)$
This is somehow connected to this question.
I can think of at least four topologies to put on $C_c(M)$:
Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney $C^\...
8
votes
2
answers
986
views
Division of Distributions by Polynomials
Let $P(z)$ be a non-null complex polynomial in $n$ variables $z=(z_1,\dots,z_n)$:
\begin{equation}
P(z)=\sum_{|\alpha| \leq N} c_{\alpha} z^{\alpha},
\end{equation}
where as usual for every $\alpha=(\...
8
votes
1
answer
1k
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Traces of Sobolev spaces
Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...
8
votes
2
answers
443
views
Which complete orthomodular lattices arise from von Neumann algebras?
Let $A$ be a von Neumann algebra. Then a classic observation is that the set of projections $\Pi(A)$ is naturally a complete orthomodular lattice.
Question 1: Is the construction $A \mapsto \Pi(A)$ a ...
8
votes
0
answers
987
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Strictly singular operators and their adjoints
This is a question I thought about a while back and figured I'd throw it out there to see if anyone has some insight that I am missing.
Let $X$ and $Y$ be infinite dimensional separable Banach ...
7
votes
1
answer
322
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Extremal problem for 2-dimensional lattices
Given a lattice $L$ in a Banach space $(B,\|\;\|)$, one denotes by $\lambda_1(L)$ the least norm of a nonzero element in $L$, and by $\lambda_k$ the least $\lambda$ such that there is a linearly ...
7
votes
1
answer
922
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Compactness of the unit ball of a Banach space for topologies finer than the weak* topology
Let $(\mathcal{X} , \|\cdot \|_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|_{\mathcal{X}'}$ the dual norm and define also the topological dual $\...
7
votes
1
answer
571
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Weak* continuity of positive parts, again
Bill Johnson pointed out to me yesterday that the map $$f \mapsto f^+ = \max(f,0)$$ is not weak* continuous on $l^\infty$. Nonetheless, I think I can prove that if $V$ is a linear subspace of $l^\...
7
votes
2
answers
2k
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Uniform bound on the eigenfunctions of the Laplacian
Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...
7
votes
4
answers
2k
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Invariant means on the integers
Let $A\subseteq\mathbb Z$, as usual we define the lower Beurling density $d^{-}(A)=\lim\inf_{n\rightarrow\infty}\frac{|A\cap[-n,n]|}{2n+1}$ and the upper Beurling density $d^+(A)=\lim\sup_{n\...
7
votes
5
answers
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Convexity of distance-to-boundary function
Let $\Omega\subset\mathbb{R}^{n}$ be an open,
bounded convex domain. Denote $d_{\Omega}:\Omega\rightarrow\mathbb{R}$
the distance-to-boundary function, that is,
$$
d_{\Omega}\left(x\right):=\inf\left\...
7
votes
1
answer
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Orthonormal bases on Reproducing Kernel Hilbert Spaces
Recall that a Hilbert space $\mathcal{H}$ is a reproducing kernel Hilbert space (RKHS) if the elements of $\mathcal{H}$ are functions on a certain set $X$ and for any $a\in X$, the linear functional $...
7
votes
1
answer
863
views
Lebesgue differentiation theorem holds on locally doubling space?
It's known that for a metric space with doubling measure $(X,\mu)$, the Lebesgue differentiation theorem holds , i.e. If $f:X\to \mathbb{R}$ is a locally integrable function, then $\mu$-a.e. points ...
6
votes
3
answers
2k
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A question on fractional derivatives
I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange.
I just wanted to ask if there is a notion of ...
6
votes
1
answer
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Prove that the flow of a divergence-free vector field is measure preserving
On page 3 of this preprint, after recalling the definition of flow generated by a vector field, the authors remark that "a necessary condition for a flow $\varphi_t(\cdot)$ generated by $a(t, \cdot)$ ...
6
votes
1
answer
219
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Quantum group representations from (convolution) matrix units?
Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on $A=F(\...
6
votes
1
answer
662
views
Is there an operator algebraic reformulation of the invariant subspace problem?
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
6
votes
1
answer
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Lipschitz function of independent subgaussian random variables
This question was asked here, but I have reason to believe that it's a serious research question appropriate for this forum (also, the answers given at the link aren't satisfactory).
If $X\in\mathbb{...
6
votes
2
answers
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Vector Fields in a Riemannian Manifold
Suppose $(M,g)$ is a Riemannian manifold.
Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator?
Thanks
6
votes
1
answer
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Covering number of Lipschitz functions
What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?
Only 2 results I have found so far are,
That the $\infty$-...
6
votes
1
answer
747
views
An equivalence relation on the space of polynomials in one complex variable
Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$
where these integers are ...
5
votes
1
answer
272
views
Regularity of the Radon transform with respect to the original function
Consider a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ (whose properties are to be specified). I note $\mathbb{S}^{d-1}$ the hypersphere and the Radon transform of $f$ defined for $(t,\theta) \...
5
votes
1
answer
315
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Are the polyhedral cones the only examples of cones that remains closed when they are added to vector subspaces?
Let $C \subset \mathbb{R}^{n}$ be a closed convex cone. If one wants to know whether the linear map $T:\mathbb{R}^{n} \to\mathbb{R}^m$ sends the closed set $C$ to another closed one, $T(C)$, it is ...
5
votes
1
answer
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Is every distribution a linear combination of Dirac deltas?
My question is whether Dirac-type distributions over an Abelian group define a basis of the Schwartz-Bruhat space $\mathcal{S}(G)^\times$ of tempered distributions on $G$, so that any distribution $f\...
5
votes
1
answer
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Inner product of linear bounded operators between Hilbert spaces
Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces.
Can we equip $L(X,Y)$ with a natural inner product? I think it should look like
$\...
5
votes
2
answers
400
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Backward heat equation and forward perturbed heat equation well posed?
I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
5
votes
1
answer
431
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Modulus of continuity of flow for non-Lipschitz vector fields satisfies Osgood condition
An Osgood modulus of continuity is an increasing function $\omega:(0,1]\to(0,1]$ such that $\int_0^1\frac{dt}{\omega(t)}=\infty$.
We say a vector field $X$ satisfies Osgood condition with modulus $\...
5
votes
0
answers
227
views
What is the smallest number of hyperplanes covering $\ell_2$?
For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$.
By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...
4
votes
2
answers
552
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Is a specific sequentially closed subset of $M([0,1])$ closed?
Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\...
4
votes
0
answers
184
views
improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
4
votes
1
answer
410
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A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $
Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}_{q}}(2) $ is defined to be the universal unital $ C^{*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the ...
4
votes
1
answer
729
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What is the dual space of $L^p$(conservative vector fields on a bounded set)?
First, some background: I wanted to prove that, if $f$ is a measurable function such that $\nabla f\in L^p_\text{loc}(\mathbb R^n)$, then $f\in L^p_\text{loc}(\mathbb R^n)$, $p\in(1,\infty)$. This is ...
4
votes
1
answer
202
views
Mapping properties of backward and forward heat equation
In a previous question on mathoverflow, I asked about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions.
The ...
4
votes
0
answers
2k
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metric entropy for Lipschitz functions
Suppose $(X,d)$ is a metric space of unit diameer and let $F$ be the collection of all $1$-Lipschitz functions mapping $X$ to $[-1,1]$, equipped with the sup-norm $||\cdot||_\infty$.
I am interested ...