All Questions
Tagged with functional-analysis or fa.functional-analysis
912 questions
15
votes
3
answers
2k
views
Can the Riemann integral be defined through a closure/completion process?
Let us consider real-valued functions on the bounded interval $[0,1]$. A "step function" means an element of the vector space spanned by indicator functions of (points and) intervals in $[0,1]$ (the ...
- 32.5k
15
votes
2
answers
3k
views
Generalizations of the Tietze extension theorem (and Lusin's theorem)
I am reasking a year-old math.stackexchange.com question asked by someone else.
(For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.)
The Tietze ...
- 6,287
15
votes
1
answer
2k
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Bases for spaces of smooth functions
Let $S$ denote the space of rapidly decreasing sequences, which means sequences $a=(a_k)_{k=1}^\infty$ such that the numbers $p_d(a)=\sup\{k^d|a_k| : 1\leq k<\infty\}$ are finite for all $d\in\...
- 56.9k
14
votes
2
answers
1k
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Are smooth functions tame?
I know the article of Hamilton on the inverse function theorem of Nash and Moser (with the same title) where he proves that $C^\infty(M)$ is a tame Fréchet space, when $M$ is closed or compact with ...
- 9,853
14
votes
6
answers
3k
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What's a natural candidate for an analytic function that interpolates the tower function?
I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that ...
- 4,466
14
votes
2
answers
6k
views
Are weak and strong convergence of sequences not equivalent?
For some infinite-dimensional Banach spaces $E$, it is easy to find sequences $\langle x_i:i\in\mathbb N_0\rangle$ which converge to zero weakly but not in the norm topology, i.e. we have $\lim_{i\to\...
- 3,584
13
votes
3
answers
1k
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Is the set of separable quantum states closed?
Let $\mathcal H,\mathcal H'$ be Hilbert spaces (not necessarily separable).
A "separable state" is a trace-class operator of the form $\sum_i \rho_i\otimes\rho_i'$ where $\rho_i,\rho_i'$ are positive ...
- 896
13
votes
0
answers
818
views
Covering number estimates for Hölder balls
Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
- 5,405
13
votes
0
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491
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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 ...
- 82.7k
12
votes
3
answers
2k
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Reference request: Simple facts about vector-valued Sobolev space
Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\...
- 29.7k
12
votes
3
answers
1k
views
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 ...
- 4,461
12
votes
1
answer
575
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 ...
- 4,878
12
votes
3
answers
2k
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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 ...
- 1,846
11
votes
4
answers
2k
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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 ...
- 16.4k
11
votes
2
answers
1k
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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 \...
- 1,773
11
votes
1
answer
258
views
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 ...
- 32.5k
11
votes
2
answers
1k
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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 ...
- 711
11
votes
4
answers
1k
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Example of noncomplete quotient of complete lcs mod closed subspace
The following statement is well-known: for a Fréchet space $V$ and a closed subspace $W \subseteq V$ the quotient $V / W$ is again complete and hence a Fréchet space. For the particular case of a ...
- 8,098
11
votes
1
answer
1k
views
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 ...
- 109k
11
votes
2
answers
714
views
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 $A_n:=I_n-M_n^2$. ...
- 43.2k
10
votes
1
answer
492
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 ...
- 2,754
10
votes
1
answer
783
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 ...
- 976
10
votes
2
answers
926
views
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 ...
- 4,705
10
votes
0
answers
761
views
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 ...
- 1,616
9
votes
1
answer
2k
views
Rate of convergence of smooth mollifiers
How does one figure out/prove the rate of convergence (in some norm) of mollifiers given a function bounded in some other norm (say Sobolev space, Besov space)? Also, is there a dimensional analysis ...
- 2,243
9
votes
5
answers
2k
views
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\...
- 91
9
votes
1
answer
4k
views
What are some characterizations of the strong and total variation convergence topologies on measures?
I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.
The Wikipedia article on convergence of measures defines three kinds of convergence: ...
- 203
9
votes
1
answer
956
views
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 ...
- 1,143
9
votes
1
answer
1k
views
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}(\...
- 28k
8
votes
2
answers
496
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 ...
- 63.9k
8
votes
1
answer
1k
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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 $\...
- 2,306
8
votes
2
answers
2k
views
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^\...
- 81
8
votes
2
answers
1k
views
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 ...
- 6,853
8
votes
0
answers
1k
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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 ...
- 1,073
8
votes
2
answers
1k
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=(\...
- 640
8
votes
1
answer
4k
views
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$-...
- 2,246
7
votes
1
answer
2k
views
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 $...
- 577
7
votes
1
answer
334
views
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 ...
- 16.6k
7
votes
1
answer
246
views
A notion of restricted injectivity for Banach spaces
I apologize in advance if this is well-known.
Let $X$ be a Banach space. Let's call only for this post that $X$ is self-injective if for every closed subspaces
\begin{equation}
A\subseteq B\subseteq X ...
- 2,605
7
votes
2
answers
2k
views
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 ...
- 73
7
votes
1
answer
606
views
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^\...
- 42.8k
7
votes
1
answer
907
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 ...
- 471
7
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 ...
user168611
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\...
- 6,034
7
votes
1
answer
754
views
Closed convex hull in infinite dimensions vs. continuous convex combinations
tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$?
I am essentially asking for the most general, infinite-dimensional analogue of ...
- 71
6
votes
2
answers
1k
views
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
- 4,135
6
votes
1
answer
1k
views
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{...
- 6,541
6
votes
1
answer
1k
views
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 $a(t, \cdot)$ ...
- 839
6
votes
1
answer
765
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 ...
- 356
6
votes
1
answer
680
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 ...
