All Questions
12,777 questions
7
votes
0
answers
1k
views
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?
- 29.1k
18
votes
3
answers
2k
views
Does Riemann map depend continuously on the domain?
I've always taken this for granted until recently:
In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations $\...
- 375
3
votes
1
answer
2k
views
Existence of solution for Poisson problem with pure Neumann BCs
Hello all,
Does the following boundary value problem admit unique solutions $q$:
$- \Delta q + \beta q = f$, $x \in \Omega$
$ \nabla q \cdot \vec{n} = g $, $x \in \Gamma := \partial \Omega$,
...
- 53
5
votes
3
answers
843
views
Topology on the set of linear subspaces
Hello,
let $X$ be a seperable Hilbert space. Let $(e_i)_i$ be a Hilbert basis, and for each index let $E_i = \langle e_1,\dots,e_i \rangle \subset X$ the span of the first $i$ basis vectors. For any $...
- 5,327
43
votes
3
answers
9k
views
Why the name 'separable' space?
It is well known that a separable space is a topological space that has a countable dense subset. I am wondering how is this related to the name 'separable'? Any intuition where the name come from?
- 1,157
1
vote
0
answers
460
views
Topology for test functions [closed]
One naive way to define a topology on test functions ${\mathcal D}(\Omega)$ would be to exhaust $\Omega$ by compacts $(K_n)$ and to take the metric induced by the semi-norm system
$$
{\| f \|} _ {n} :=...
- 53
1
vote
0
answers
335
views
Universally open morphism with reduced fibers.
Hi.
I asked in the last post if, for a flat morphism $f:X\rightarrow S$ of complex spaces with reduced fibers and $S$ reduced, $X$ is reduced or not. In the algebraic setting, Liu said that the ...
- 435
37
votes
2
answers
2k
views
Moving one family of commuting self-adjoint operators to another without losing commutativity on the way
This is actually not a question of mine, so I'll be short on motivation and say nothing beyond that if this were true, a few fancy harmonic analysis techniques that a colleague of mine used in proving ...
- 61.9k
2
votes
1
answer
373
views
Strong measurability reference
I'm reading a book on Lyapunov Exponents by Lian and Lu in which they refer to strong measurability of operator-valued maps. They define this by saying an operator valued map $T:\Omega\to L(X,X)$ is ...
- 23.2k
11
votes
0
answers
309
views
Combinatorial Hilbert spaces
Any closed subspace $V\subset {\ell}^2(\omega)$ has associated to it a subset ${\cal S}_V$ of ${\cal P}(\omega)$, call it a combinatorial Hilbert space, namely the set of all supports of all vectors ...
- 17.6k
22
votes
13
answers
7k
views
Is there a "crash-course" book on Abelian varieties (e.g., an introduction for physicists)?
Hello,
In our (rather applied) theoretical physics research, we have encountered an important class of problems, which seem to require an understanding of Abelian functions (unfortunately, this ...
- 649
6
votes
1
answer
1k
views
Zeros of a holomorphic function
Suppose Ω is a bounded domain in the plane whose boundary consist of m+1 disjoint analytic simple closed curves.
Let f be holomorphic and nonconstant on a neighborhood of the closure of Ω such that
|...
- 101
12
votes
1
answer
859
views
Who first found this characterization of Lebesgue integration?
Write $L^1$ for the Banach space $L^1([0, 1])$. Given $f \in L^1$, define $f_1, f_2 \in L^1$ by
$$
f_1(x) = f(x/2),
\qquad
f_2(x) = f((x + 1)/2).
$$
Let $I = \int_0^1$. Then $I$ is the unique ...
- 27.7k
5
votes
2
answers
1k
views
Flat map with reduced fibers.
Hi.
Let $f:X\rightarrow S$ be a flat, surjective morphism of complex spaces with reduced fibers over $S$ reduced.
Q1: Is $X$ reduced too?
Q2: Is the property " reduced fiber" preserved by base ...
- 435
9
votes
1
answer
972
views
Non-standard enlargements, $\zeta(s)$ and analytic continuation
Consider an extension of the Riemann zeta function $\zeta(s)$ where $s$ now runs over a non-standard enlargement of the complex plane.
Observe that if $s=\sigma + it$ with $\sigma>1$ real and ...
- 17.6k
16
votes
3
answers
5k
views
Integration of differential forms using measure theory?
Setup: Let $(M,g)$ be a (possibly non-compact) Riemannian manifold with volume density $d_gV$. Then one may think of $(M,g)$ as a measure space $(\Omega,\mathcal{A},\mu)$, where $\Omega:=M$, $\mathcal{...
- 408
32
votes
0
answers
6k
views
A paper to the question, if the six dimensional sphere is a complex manifold [duplicate]
for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf
Because I am not able to ...
- 408
2
votes
0
answers
245
views
Dimension of pluripolar sets
Let $\Omega$ be an open set in $\mathbb C^n$, and let $A$ be a closed pluripolar set in $\Omega$. Is there a notion of dimension of $A$ such that the following theorem is true?
Theorem.
Let $\phi$ ...
4
votes
1
answer
1k
views
Irreducibility of Analytic Sets
How does one prove that an Analytic set $V$ in $C^n$ is irreducible if the set of regular points $V^*$ is connected?
Proceeding by contradiction, if we assume that $V$ is in fact reducible and if $...
- 537
1
vote
1
answer
908
views
About the exponential bounds for modified Bessel function
Dear colleagues,
I want to know if there are some results on the bounds of modified Bessel functions $I_\alpha(x)$ and $K_\alpha(x)$? Especially, I need the exponential bounds for them, that is to ...
- 35
1
vote
1
answer
462
views
Derivative of functional
Define $J\colon W_0^{1,2}(\Omega)\to \mathbb{R}$ by $J(u)=\int_\Omega u^{p+1}dx$ for $p\in (1,\frac{n+2}{n-2})$.
Is $J'(u)(v)=\int_\Omega (p+1)u^pvdx$?
- 13
2
votes
0
answers
390
views
Boundary behavior of Kähler cone with curvature restriction
Let $(M,\omega)$ be a compact Kähler manifold. The boundary behavior of Kähler cone is very interesting; however,it's hard to understand.
A fundamental result is due to Demailly and Paun: they ...
- 247
-1
votes
1
answer
1k
views
Kolodziej's acta paper "the complex monge-ampere equation"——a detailed ploblem [closed]
Recently,I am reading kolodziej's acta paper,there are some ditails that i do not know clearly.
In the top line of page 99,"it's no restriction to assume that for each s we have $\nu(\cup_{I\in{B_s}}...
- 247
3
votes
3
answers
1k
views
Pedagogical question concerning $\Gamma(z)$
Pedagogically speaking, I see two problems with defining
$\Gamma(z)$ (at least for real $z$) by the limit
$$\Gamma(z)=\lim_{m\to\infty}\frac{m! m^z}{\prod_{i=0}^m (z+i)}$$
as compared with the formula
...
- 17.6k
5
votes
0
answers
584
views
Studying primes via the gamma function alone: $(x+1)\prod_n \Gamma(\frac{x}{n}+1)^{\mu(n)}$
Various questions on MO concerning the "surprise" occurrence of the gamma function in the functional equation of the Riemann zeta function got me wondering whether the Gamma function alone suffice for ...
- 17.6k
34
votes
2
answers
3k
views
Can we recover a von Neumann algebra from its predual?
By definition, a von Neumann algebra is a C*‑algebra A
that admits a predual, i.e., a Banach space Z such that
Z* is isomorphic to the underlying Banach space of A.
(We require that isomorphisms in ...
- 37.8k
8
votes
2
answers
470
views
Analytic functions with isotopic x-rays
Following Arias-De-Reyna, the x-ray of an analytic function $f$ means markings on the complex plane, with one color showing the real locus of $f$ and another color the purely imaginary locus.
...
- 17.6k
49
votes
4
answers
6k
views
If the Riemann Hypothesis fails, must it fail infinitely often?
That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. ...
- 17.6k
8
votes
4
answers
888
views
$\ell^p$ version of singular values
I am embarrassed to pose this question. It is a generalization of a question asked less than 24 hours ago by an unknown (Google), which has been deleted since then, presumably by its author themself.
...
- 52.3k
13
votes
1
answer
2k
views
Surgery in complex geometry
I've been thinking about surgery on complex manifolds. Not very seriously, but just to the point that I think it's odd how there's almost no mention of it in the literature. I figure there's something ...
- 4,343
11
votes
1
answer
806
views
Algebraicity of Eigenvectors in a Hilbert space
Let $(e_j)_{j\in\mathbb N}$ be an orthonormal basis of a Hilbert space $V$. Let $T:V\to V$ be continuous, selfadjoint linear operator.
Assume that for all $i,j\in\mathbb N$ the number $\langle Te_i,...
user1688
9
votes
1
answer
2k
views
The Invariant Subspace Problem: examples
Question. Is there a concrete example of a bounded linear operator on a Hilbert space for which it is not known if it has a non-trivial closed invariant subspace?
[Added 24.01.2011: According to ...
- 22.3k
2
votes
2
answers
867
views
Decomposition of an abelian von Neumann algebra
Hi, I came across the statement below and I couldn't figure out why it is true. I was hoping someone could explain it or give me a good reference. Thank you in advance.
"Let $\pi$ be a non-degenerate ...
14
votes
2
answers
926
views
"Explicit" embedding of $\ell^1$ as a closed subalgebra of a direct sum of matrix algebras
For sake of brevity let $A$ denote the Banach algebra formed by equipping $\ell^1({\mathbb N})$ with pointwise multiplication. This algebra is clearly not isomorphic as a Banach algebra to any uniform ...
- 25.8k
3
votes
0
answers
207
views
plurisubharmonic sublevel sets
Let $X$ be a complex manifold, let $\Omega \subseteq {\bf C} \times X$ be defined by
$\Omega = \{ (z,p) \in {\bf C} \times X : a(p) < Im z < - b(p) \} $ where $a$ and $b$ are plurisubharmonic ...
6
votes
1
answer
727
views
Is this method of "fractional sums" using a Fourier series viable?
Hi.
I have this idea about developing what I call a "continuum sum", that is, a method to "add up a non-integer number of terms", i.e. to see if there is a "natural" way to assign a meaning to the ...
- 1,687
68
votes
1
answer
13k
views
Behaviour of power series on their circle of convergence
I asked myself the following question while preparing a course on power series for 2nd year students. Let $F$ be the set of power series with convergence radius equal to $1$. What subsets $S$ of the ...
- 683
15
votes
5
answers
680
views
Idiosyncratic characterizations of $\ell^p$, for $p\not=1,2,\infty$
Do there exist, either in the literature or in folklore, theorems
that characterize some particular $\ell^p$ space(s) ($p\not=1,2,\infty$)?
Such a theorem should reveal the particular space(s) as ...
- 17.6k
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
39
votes
3
answers
14k
views
Is the Invariant Subspace Problem interesting?
There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the Invariant Subspace Problem, he says (paraphrasing) "...this question is still open. It is also an ...
- 1,241
12
votes
2
answers
1k
views
Landau's constant
(Hi. This is my first question here.)
A well known result in complex analysis says that there is an $\varepsilon\gt 0$ such that if $f$ is holomorphic in (a neighborhood of) the closed disk ${\mathbb ...
- 421
11
votes
1
answer
800
views
on common fixed points of commuting polynomials (and rational functions)
By the Ritt's classification, for any pair of commuting polynomials (i.e. $f(g(z))=g(f(z))$) over $\mathbb C$ there is a common fixed point of them. My questions are:
Is that true that this can be ...
- 1,422
7
votes
2
answers
1k
views
Relationships between the roots of an entire function and the roots of its derivative
Hey everyone,
I would like to know if anybody could help me find references for the following.
Take a suitably well defined entire function $f(x)$ and it's derivative $\tilde{f}(x)$ to which the ...
- 211
2
votes
0
answers
800
views
Controlling the Lipschitz norm of the limit of a sequence of functions
Consider the Fréchet space $\Omega = C(\mathbb R^d)$ of real-valued continuous functions equipped with the seminorms $$\|f\|_D := \sup_{x,y \in D} \left\{ |f(x)|, \tfrac{|f(x)-f(y)|}{|x-y|} \right\}, \...
- 8,512
4
votes
1
answer
720
views
Are coordinate functionals on complete vector spaces always continuous?
(I'm just adding the completeness condition to $V$ from this 2 month old question of mine, because I realized it's relevant to whether Bill Johnson's answer to this 4 month old question of mine ...
user5810
8
votes
1
answer
678
views
Spectral theory of pseudo-differential operators
Consider a finite rank complex bundle $E$ over $S^1$ with connection $\nabla$. Let $Q_0, Q_1 \in C^\infty(S^1, E)$ be pseudo-differential operators. $Q_0$ is defined by the symbol $\sigma_0(x, \xi) =...
3
votes
1
answer
842
views
An elementary introduction of Colombeau's generalized function theory
Hello, I am wondering whether anyone know an elementary reference for Colombeau's theory on the multiplication of distributions? I encountered the problem of the square of Delta function. I need a ...
- 1,649
2
votes
2
answers
528
views
Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution
Let $G$ be a discrete group.
Do you know characterizations of amenable groups which use the space $\ell_1(G)$ and convolution?
I only know Johnson's theorem:
A group is amenable if and only if the ...
6
votes
1
answer
1k
views
Harmonic forms on Ricci-flat Kahler manifolds
Let $X$ be a compact Kahler manifold with $c_1(X) = 0$. Any Kahler metric $\omega$ on $X$ gives a Laplacian $\Delta_\omega$ and the $(1,1)$-form $\omega$ is harmonic with respect to this Laplacian.
...
- 4,343
7
votes
3
answers
1k
views
Compactness properties of plurisubharmonic functions
I'm quite interested in this topic, but the main text on Several Complex Variables say little of nothing about it. Here are my questions, and I'd be grateful of any reference or information.
Let $\...
- 60.5k