All Questions
12,776 questions
4
votes
2
answers
4k
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Proof of Young's convolutions inequality for a general measure on $\mathbb R^d$
Is Young's inequality true for an arbitrary measure on $\mathbb R^d$? If so, where can I find a proof of it? In particular, where can I find the proof of the discrete version (i.e the version for $\...
- 2,745
4
votes
2
answers
627
views
The link of a singular quintic hypersurface in CP^4
Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by
$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$
we get a singular variety for $\epsilon=0$ with 125 singular points.
I know ...
- 93
15
votes
2
answers
1k
views
Asymptotic approximation of $x^\alpha$ by entire functions
Given a non-integral real $\alpha$, is there an entire (see http://en.wikipedia.org/wiki/Entire_function) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$
for $x\rightarrow+\infty$ (with $...
- 17.9k
14
votes
3
answers
3k
views
Analytic continuation of holomorphic functions
Analytic/meromorphic continuation is a difficult problem in general. For "motivic L-functions", the idea of proving their analytic continuations by first proving their modularity goes back, I guess, ...
- 4,265
-4
votes
1
answer
514
views
Meaning of the Mobius transformations video [closed]
What is this video trying to tell us?
http://www.youtube.com/watch?v=JX3VmDgiFnY
The statement that fractional linear transformations correspond to rotations of the sphere under the stereographic ...
- 1,783
12
votes
3
answers
2k
views
Relevance of the complex structure of a function algebra for capturing the topology on a space.
This question is the outcome of a few naive thoughts, without reading the proof of Gelfand-Neumark theorem.
Given a compact Hausdorff space $X$, the algebra of complex continuous functions on it is ...
- 3,699
15
votes
4
answers
2k
views
Naive questions about "matrices" representing endomorphisms of Hilbert spaces.
This is a very basic question and might be way too easy for MO. I am learning analysis in a very backwards way. This is a question about complex Hilbert spaces but here's how I came to it: I have in ...
- 41.4k
7
votes
3
answers
1k
views
Can Stein's maximal principle be strengthened?
Let $T$ be an operator on $S(G)$ where $G$ is the line $R$ or the circle $T$, and $S(G)$ denotes the Schwartz space of functions on $G$.
We can ask if the operator T is bounded (as an operator from $...
- 13k
22
votes
2
answers
2k
views
Examples of loss of regularity by "creation of topology"
I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem considered)...
- 2,041
27
votes
4
answers
3k
views
Genealogy of the Lagrange inversion theorem
A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, ...
- 60.5k
21
votes
5
answers
18k
views
When is Sobolev space a subset of the continuous functions?
If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L_2(\Omega): \frac{\partial w}{\partial x_i}\in L_2(\Omega), i=1,...,d)$. My tutor has repeated several ...
- 3,217
30
votes
3
answers
4k
views
What is special about polylogarithms that leads to so many interesting identities and applications?
I have heard that Polylogarithms are very interesting things. The wikipedia page shows a lot of interesting identities. These functions are indeed supposed to have caught the attention of Ramanujan. ...
- 3,699
1
vote
0
answers
133
views
Square powers of hemicontinuous operators
Let H be an infinite dimensional real Hilbert space.
A [not necessarily linear] mapping of H into itself is said to be hemicontinuous if it is continuous from each line
segment of H to the weak ...
- 4,060
5
votes
1
answer
7k
views
Dual Spaces of Sobolev Spaces
I will consider Sobolev spaces with $p=2$, only, so that they are Hilbert spaces. Hence the Sobolev inner product identifies each Sobolev space with its dual. In other words, I have an isomorphism $H^...
- 807
6
votes
0
answers
161
views
Multiplicity of zero (higher dimensional analog)
Consider a sistem of n holomorphic equations with n unknowns in a neighborhood of zero. Suppose that a solution in a neighborhood of 0 is a k-dimensional manifold.
I want to associate to it some ...
- 61
2
votes
2
answers
768
views
Elementary vector measure question: what am I doing wrong?
This is an edited post of a post I made on sci.math (e.g. to fit MO markup) with
an elementary question on vector measures. Since it is almost a week and I have
received no answers, I am trying here. ...
- 1,848
3
votes
1
answer
556
views
"Radon-Nikodym theorem" for nonabsolute continuous measures
Recently, in a particular problem I was solving, I needed some kind of Radon-Nikodym theorem for measures where one of them is not necessarily absolutely continuous with respect to other.
My colleague ...
- 349
7
votes
1
answer
286
views
a.e. convergence of the powers of an operator built from rotations
Consider two numbers $a,b\in R/Z$ and some integer $p\geq 1$. Let $T:L^p(R/Z)\rightarrow L^p(R/Z)$ be the operator given by
$$T(f)(x)=1/2(f(x+a)+f(x+b))$$
For which values of $a,b$ do we have almost ...
- 18.7k
28
votes
6
answers
12k
views
Almost orthogonal vectors
This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
- 18.7k
94
votes
1
answer
11k
views
The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
- 23.5k
10
votes
2
answers
1k
views
Are operators with trivial spectrum nilpotent in a sense?
Being far from analysis, I recently learned about the Invariant subspace problem and came up with the following (perhaps simple or well-known) question.
Let $H$ be a separable complex Hilbert space ...
- 32.4k
9
votes
1
answer
893
views
Perturbations of an operator that disconnect the spectrum
The following question came to me while working on a technical matter about transversality in infinite dimension, and I'm really curious to know whether it has an affirmative answer at least under ...
- 60.5k
7
votes
1
answer
1k
views
Banach spaces with a certain separability property
In Ledoux and Talagrand's "Probability in Banach Spaces", for technical reasons they frequently assume that a Banach space $B$ has the property that the unit ball of $B^*$ contains a countable subset $...
- 11.4k
1
vote
2
answers
3k
views
unit sphere is weak dense in the unit ball
As I remember the following is true:
Fact: for every infinite-dimensional normed space $X$
the unit sphere $S$ is weak-dense in the unit ball $B$.
Please help me find a reference.
Thanks in ...
- 51
11
votes
2
answers
2k
views
What's wrong with compact-open topology on the space of maps?
Given a smooth vector bundle $E$ with non-compact base, let
$\Gamma(E)$ be the space of $C^\infty$ sections equipped with compact-open $C^\infty$-topology.
I have heard that $\Gamma(E)$ is not ...
- 29.1k
2
votes
2
answers
801
views
Domains of holomorphy in the complex plane
There is a proof of Mittag-Leffler's theorem with an explicit construction of a holomorphic function with the prescribed poles with prescribed order and residues, for a countable discrete set of ...
- 3,699
8
votes
1
answer
1k
views
Borel(X) = \sigma(X') for X non-separable
Let $X$ be a Banach space, $X' = \mathcal{L}(X, \mathbb{K})$ its dual space. Denote by $\mathcal{B}(X)$ the $\sigma$-algebra of Borel sets and denote by $\sigma(X')$ the $\sigma$-algebra which is ...
- 1,338
6
votes
2
answers
1k
views
Quantitative questions about the size of a finite epsilon net
Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of $\epsilon$...
- 943
1
vote
2
answers
534
views
Local representation of an analytic sets
Let V be a analytic set of $C^n$, $I(V)$ is the sheaf of ideals of V (the sheaf whose stalks are ideals defining germs of V at its points). Since $I(V)$ is a coherent analytic sheaf, we see that in a ...
- 750
7
votes
1
answer
2k
views
Current status of Bloch constant and Landau constant bounds
The Bloch constant B (based on a theorem introduced by André Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance ...
- 948
4
votes
1
answer
985
views
weak convergence in infinite dimensional spaces
Weak convergence can be tricky when dealing with infinite dimensional spaces. For example, the usual Levy's continuity theorem does not extend readily to separable Banach spaces.
Consider a (...
- 2,133
19
votes
5
answers
16k
views
What does "kernel" mean in integral kernel?
In functional analysis, there is the term "integral kernel". Examples are Possion kernel, Dirichlet kernel etc.
In algebra, the term kernel of a homomorphism refers to the inverse image of the zero ...
user2529
6
votes
0
answers
2k
views
Weak lower semi-continuity
Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type
$F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will even ...
4
votes
2
answers
442
views
Elementary functions with zeros only at the positive integers
Does there exist a (meromorphic) elementary function $f(z)$ that is zero at all the positive integers $z = 1, 2, 3, \ldots$ and only at those points?
Edit: an elementary function can be written as a ...
- 2,235
12
votes
3
answers
1k
views
Drawing conclusions by NOT using AC.
The existence of non-measurable subsets and functions on $\mathbb{R}$ require the use of the axiom of choice. That is, there exist models of ZF in which all subsets of (and hence all functions defined ...
- 4,790
3
votes
2
answers
618
views
Schwarz Lemma in terms of conformal surfaces or holomorphic curves?
Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting.
Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...
user2529
7
votes
2
answers
2k
views
What is the "Krein-Milman theorem for cones"?
Update: The question is completely answered. I had overlooked a reduction to the self-adjoint case, and the latter can be proved using a Hahn-Banach separation theorem. Thanks to Matthew Daws for ...
- 7,329
7
votes
2
answers
808
views
Is a subspace with a certain property dense in the dual of a vector space?
Suppose we have a normed vector space $V$ and its dual $V^*$, and suppose that $X \subseteq V^*$ has the property that for every $v \in V$, there is some $\phi \in X$ with $\Vert \phi \Vert = 1$ such ...
- 73
11
votes
1
answer
654
views
Nonseparable Hilbert spaces as quotients of spaces of bounded functions
Is the following result true: the Hilbert space $\ell^{2}\left(2^{\Gamma}\right)$ is a quotient of $\ell^{\infty}\left(\Gamma\right)$ for any
uncountable $\Gamma$ ? [I think it is, but cannot remember ...
- 4,060
7
votes
1
answer
347
views
Nonexistence of determinantal functional equation for $\arccos$
Suppose I have distinct real numbers $a_i \in [-1,1]$, $i \in [k]$. I want to choose real numbers $b_j, j\in [k]$ such that the matrix $(\arccos(a_i b_j))_{i,j \in [k]}$ is nonsingular.
Is this ...
- 1,021
0
votes
1
answer
635
views
Topological dual and the notions of "smaller" and "larger" than...
Hi,
I've read this sentence but I can not understand what it means
[...] $\Phi'$ is the topological dual of some dense space $\Phi$ of $H_{aux}$ [...] Notice that the choice of $\Phi$ is subject to ...
- 733
5
votes
1
answer
403
views
Local form of a real-analytic function taking values in a Banach space
Let $B$ be an infinite-dimensional Banach space, and let $M\subset\mathbb{R}^n$ be a neighborhood of the origin in $\mathbb{R}^n$.
Suppose that $I:M\to B$ is a real-analytic function with $I(0)=0$ ...
- 6,548
6
votes
2
answers
4k
views
Bounded and weakly bounded sets in top. vector spaces
Consider a locally convex topological vector space V over the complex numbers. Is it true that every weakly bounded subset of V is indeed bounded? If not, what additional requirements are needed for ...
- 61
3
votes
2
answers
766
views
Borel vs measure for all Borel measures
Let X be locally compact and Hausdorff, and let $f:X\rightarrow\mathbb R$ be a function. Suppose that for all finite regular (positive) Borel measures $\mu$, we know that $f$ is $\mu$-measurable. ...
- 18.7k
11
votes
2
answers
2k
views
Complex analytic vs algebraic families of manifolds
I'm studying the deformation theory of compact complex manifolds as developed by Kodaira and Spencer. On the side I'm reading as much about deformation theory in general as I can get my hands on (and ...
- 4,343
5
votes
4
answers
1k
views
Is any continuous linear operator from a dual Banach space to a separable Hilbert space the strong-operator limit of a net of adjoint operators of less or equal norm ?
Let $E$ be an arbitrary Banach space and let $T:E^{*}\rightarrow\ell^{2}$
be a linear continuous operator. Is it true that $T$ must be the
$so$-limit (i.e., limit w.r.t. the strong operator topology) ...
- 4,060
4
votes
0
answers
487
views
Convolutions and Toeplitz Operators
Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.
Let be $...
- 2,044
7
votes
0
answers
4k
views
Explicit element of $(\ell^{\infty})^* - \ell^1$? [duplicate]
Possible Duplicate:
What’s an example of a space that needs the Hahn-Banach Theorem?
It is well known that the dual of $\ell^{\infty}$ properly contains $\ell^1$ (over $\mathbb{N}$, say). ...
- 25.6k
2
votes
2
answers
354
views
A bound on linear functionals over cotype 2 spaces
This is a modification of the somewhat naive question that I asked below.
Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\...
- 2,151
15
votes
3
answers
1k
views
Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?
The following describes an extension of the Descartes Rule of Signs to polynomials with complex coefficients.
First, I need to define the notion of a "sweep"... Given a complex polynomial p(z) := c0 ...
- 1,230