All Questions
12,823 questions
6
votes
2
answers
1k
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Closed, complemented subspaces of $l^1(X)$ when $X$ is uncountable
... are all isomorphic to $l^1$ on some other index set. At least, that much I "know" from 2nd-hand sources, since the original proof is apparently in a paper of Köthe from the 1930s 1960s (in ...
- 25.8k
65
votes
14
answers
6k
views
Notions of convergence not corresponding to topologies
This question concerns the ramifications of the following interesting problem that
appeared on Ed Nelson's final exam on Functional Analysis some years ago:
Exam question: Is there a metric on the ...
- 801
0
votes
1
answer
166
views
Transformation from domains to half-spaces
In a paper I read, an elliptic boundary value problem
on a bounded domain D x (0,T) is solved by first transforming
it in a set of equations on half-spaces R^n and then applying
partial Fourier ...
45
votes
7
answers
9k
views
What's an example of a space that needs the Hahn-Banach Theorem?
The Hahn-Banach theorem is rightly seen as one of the Big Theorems in functional analysis. Indeed, it can be said to be where functional analysis really starts. But as it's one of those "there ...
- 26.8k
20
votes
3
answers
4k
views
Basis of l^infinity
Is it possible to exhibit a (Hamel) basis for the vector space l^infinity, given by the bounded sequences of real numbers?
- 1,638
6
votes
7
answers
991
views
Are there Generalisations of a Limit (for Just-divergent Sequences)?
There are certain sequences such as
0, 1, 0, 1, 0, 1, 0, 1, ...
that do not converge, but that may be assigned a generalised limit. Such a sequence is said to diverge, although in this case a phrase ...
- 524
5
votes
1
answer
1k
views
Are smooth functions on an uncountable sum continuous?
Consider the linear space $\sum_{\mathbb{R}} \mathbb{R}$. As in the Frolicher-Kriegl-Michor view, we make this into a Frolicher space as follows.
Equip it with the locally convex topology of the ...
- 26.8k
15
votes
4
answers
2k
views
Can one do without Riesz Representation?
In more detail, can one establish that the continuous linear dual of a Hilbert space is again a Hilbert space without appealing to the Riesz Representation Theorem?
For me, the Riesz Representation ...
- 26.8k
9
votes
1
answer
1k
views
"Vector bundle" with non-smoothly varying transition functions
I'm working my way through Lang's Fundamentals of Differential Geometry, and when he introduces vector bundles, he states that for finite dimensional bundles, the third axiom is redundant. I'm hoping ...
- 787
3
votes
0
answers
383
views
Neglect of Compact Quantum Metric Spaces [closed]
Does anyone have an opinion on Rieffel's theory of compact quantum metric spaces? To me it seems to be a very interesting new area of mathematics. It shows how to generalise complicated geometric ...
- 1,462
15
votes
1
answer
1k
views
Gelfand-Naimark from the category-theoretic point of view
I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative $C^*$-algebra (with unit) $\mathcal{A}$ and the $C^*$ -algebra of continuous complex-valued ...
- 2,479
1
vote
3
answers
2k
views
Various Cartan's Lemmata
I am a bit amazed by "Cartan's Lemma".. I have so far seen it in :
Algebraic Geometry sources:
Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...
- 2,275
47
votes
9
answers
6k
views
What examples of distributions should I keep in mind?
I'm learning a bit about the theory of distributions. What examples of distributions will help me develop good intuition?
Definitions: Let $U$ be an open subset of $\mathbb{R}^n$. Write $C_c^\infty(...
12
votes
3
answers
2k
views
level sets of multivariate polynomials
Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$...
- 621
5
votes
3
answers
753
views
Regularity of sparse Fourier transforms
Suppose $F$ has discrete Fourier transform $(a_n)$ where $a_n=0$ unless $n=2^k$ for some $k > 0$, in which case $a_n=1/k$ (or $a_n=1/k^2$ if you want: I'm happy with anything polynomial). What ...
- 18.7k
12
votes
2
answers
5k
views
Where was/is Compensated Compactness used?
This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
- 261
2
votes
3
answers
1k
views
Sobolev norms of eigenfunctions
Let D be a domain in R^n, and let f be an eigenfunction of the Laplacian with Dirichlet boundary condition with eigenvalue $\lambda$. Assume that f has L^2 norm 1. I want to know if I can say anything ...
21
votes
2
answers
1k
views
Is there an L^p tauberian theorem?
From Wiener's tauberian theorem we know that linear combinations of translates of f \in L^1(R) are dense in L^1(R) if and only if the Fourier transform of f never vanishes. It is also known that ...
- 13k
2
votes
3
answers
3k
views
What function has fourier series the harmonic series?
I know that this is on the boundaries of what's allowed, but hopefully someone'll answer before it gets closed!
What (periodic) function has Fourier series the harmonic series? I really want the ...
- 26.8k
3
votes
3
answers
2k
views
Error analysis of implicit functions
I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms):
$$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial f}{...
- 131
38
votes
26
answers
57k
views
Text for an introductory Real Analysis course.
Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
1
vote
6
answers
2k
views
Derivate Bessel Function with respect to order
Is there any known formulae for the derivative of the Bessel function with respect to the order of the Bessel function?
- 11
13
votes
2
answers
862
views
Motivation for BMO
At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg ...
- 261
2
votes
2
answers
1k
views
Decoupling lemma for the Lambda(p) problem
I'm attempting to work through Bourgain's paper "Bounded orthogonal systems and the $\Lambda(p)$-set problem". There is a step in the proof of the decoupling lemma that I am stuck on, and thought ...
- 13k
2
votes
2
answers
623
views
What functions are not represented by their power series?
Some functions are not represented by their power series even when they are continuous and have all the necessary derivatives. What's the best characterization of these functions? Explanations at any ...
- 3,613
72
votes
9
answers
16k
views
Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)
Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
- 1,093
3
votes
1
answer
408
views
How to do asymptotics for integrals?
What's a good way to find how fast the integral of a function is growing near a pole of the function? Here is what I mean on an example.
Look at 1/z.
If I want to find out how fast ∫0a 1/(z-&...
3
votes
3
answers
1k
views
Minimizing a functional
I have wondered the problem in http://www.helsinki.fi/~hmkokko/Stuff/Esdale/index.html for over year without success. If we try to minimize the functional equation
T(\theta ) = \int_0^L\frac {dx}{v_0\...
3
votes
1
answer
270
views
L^p Idempotent multipliers in 2 dimensions
This question is suggested by that about L^p multipliers (and the answer by Michael Lacey in particular).
Let E be a measurable set in the plane and XiE its characteristic function. We say E is an Lp ...
- 2,479
4
votes
3
answers
451
views
uniformity for Banach algebras - a three space property?
Let $A$ be a commutative, unital Banach algebra and $I \subset A$ an ideal such that $I$ with the relative norm is a uniform Banach algebra and $A / I$ with the quotient norm is uniform as well.
Does ...
- 1,338
14
votes
1
answer
3k
views
spiral of Theodorus
A long time ago when I was in college I read about making a spiral out of right triangles with sides 1 and $\sqrt{N}$. (A google search seems to indicate that this is called the Spiral of Theodorus.)
...
- 663
6
votes
3
answers
4k
views
Rigorous definition, detection and test for trending vs. mean-reverting behaviour of stochastic processes
This is a question that has haunted me for some time. In the domain of time series you always talk about trends and mean reversion. But at least to me these concepts are either defined axiomaticly ...
- 5,935
2
votes
3
answers
564
views
CAS for finding closed form solutions to PDEs and SDEs?
Are there any specialized Computer Algebra Systems (or packages for these) for finding closed form solutions to
a) partial differential equations,
b) stochastic differential equations?
If yes, what ...
- 5,935
1
vote
2
answers
479
views
"Misbehaved" differential equations
I have always been fascinated by the so called taxicab geometry first considered by Hermann Minkowski. The metric that has to be used here is a L1 distance which e.g. means that the lenght of the ...
- 5,935
26
votes
2
answers
2k
views
When is a locally convex topological vector space normal or paracompact?
All locally convex topological vector spaces (LCTVS) are completely regular, since their topology is given by a family of semi-norms. I'm interested in conditions that imply that a LCTVS is ...
- 26.8k
23
votes
4
answers
5k
views
Are proper linear subspaces of Banach spaces always meager?
Let X be a Banach space, and let Y be a proper non-meager linear subspace of X. If Y is not dense in X, then it is easy to see that the closure of Y has empty interior, contradicting Y being non-...
- 353
33
votes
6
answers
2k
views
Is there a topology on growth rates of functions?
I've often idly wondered one can say about the collection of "growth rates". By growth rate, let's say we mean an equivalence class of functions $(0,\infty) \to (0,\infty)$, where two ...
- 793
10
votes
5
answers
1k
views
What is a rigorous statement for "linear time-invariant systems can be represented as convolutions"?
In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of ...
- 2,745
25
votes
4
answers
4k
views
Which sequences can be extended to analytic functions? (e. g., Ackermann's function)
Let $\{a_n\}$ be a sequence of complex numbers indexed by the positive integers. Does there always exist an analytic function $f$ such that $f(n) = \{a_n\}$ for $n=1,2,...$? If not, are there any ...
- 5,992
11
votes
2
answers
932
views
A group action of the Heisenberg group with special symmetries
Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...
3
votes
1
answer
440
views
Asymptotics for Christoffel number
What is the order for the following sum $\sum\limits_{i=1}^{n} \frac{\lambda _i}{1-x _ i}$ where $\lambda _i$-i-th Christoffel number and $x _i$- i-th zero of n-th Legendre polynomial.
P.S
...
- 131
9
votes
5
answers
870
views
Abelianization of GL(H)
This is related to Theo's question about the abelianizations of finite dimensionsal Lie groups.
I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-...
- 629
7
votes
2
answers
1k
views
Abelianization of Lie groups
If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...
- 54.6k
9
votes
1
answer
611
views
opposite Banach space
I heard this from Haskell Rosenthal many years ago.
If V is a complex vector space, say the opposite of V is the complex vector space with the same elements, the same operations except switch scalar ...
- 41.1k
2
votes
1
answer
289
views
ODE system question
Consider a system of the form: dx/dt = f(x,y) , dy/dt=g(x,y), with the property that the associated ODE dy/dx = g(x,y)/f(x,y) has a unique solution to IVP y(0)=0.
Also, f(x,y) is smooth every except ...
- 155
12
votes
3
answers
530
views
Making an l_2 distance out of l_1 distance
If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell.
Making the grid finer doesn't ...
- 4,462
21
votes
2
answers
2k
views
In a Banach algebra, do ab and ba have almost the same exponential spectrum?
Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is ...
- 2,154
8
votes
2
answers
549
views
Where can I learn about (the asymptotics of) Toeplitz operators?
Toeplitz operators provide a natural language with which to do geometric quantization. I don't want to really understand them, and I don't need them in full generality. I'm looking for some ...
- 54.6k
3
votes
1
answer
2k
views
Hilbert Space as direct sum of subspaces with cyclic vectors
Ok,so this should be easy, however I havent taken functional analysis for a while. But given a compact self-adjoint operator on a hilbert space H(over the complex numbers), we define v to be a cyclic ...
- 31
9
votes
1
answer
395
views
Is there a coalgebraic characterisation of the hyperfinite II_1 factor?
Peter Freyd showed that the real interval [0, 1] is a final coalgebra for a functor on sets equipped with two points, which sends such a set to the 'wedge' of two copies of itself, identifying the ...
- 5,094