All Questions
10,049 questions
3
votes
3
answers
2k
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Conditional expectation of convolution product equals..
Let $X, Y$ be two $L^1$ random variables on the probablity space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra.
Consider the conditional expectation ...
- 47
9
votes
1
answer
708
views
Hilbert spaces are induced by a bilinear form. How about n-linear forms?
A Hilbert space is a complete vector space equipped with scalar product, i.e. a symmetric positive definite bilinear form.
What if we replace 'bilinear' by 'n-linear'? One might wonder, whether the $...
- 5,327
2
votes
3
answers
946
views
How can I measure the Morse index in infinite dimensions?
Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...
- 54.6k
2
votes
4
answers
3k
views
Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 54.6k
4
votes
3
answers
609
views
When is $A : C(X) \to C(Y)$ a composition operator?
A composition operator $C\_T : C(X) \to C(Y)$ with $T \in C(Y, X)$ is defined by $C\_T f := f \circ T, f \in C(X)$.
I read in the book about Composition Operators by Singh and others that a ...
- 1,338
15
votes
2
answers
2k
views
What is a projective space?
Is there a "recognition principle" for projective spaces?
What categories are there with projective spaces for objects?
Background: Although the title is a nod to What is a metric space?, ...
- 26.8k
21
votes
1
answer
1k
views
Is Dependent Choice all we really need?
http://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Is DC sufficient for the understanding of objects that are countable in some suitable sense?
For example, is DC sufficient for the full ...
- 1,199
11
votes
2
answers
862
views
Monotone Lipschitz embedding ?
In 1974, Aharoni proved that every separable metric space (X, d) is Lipschitz isomorphic to a subset of the Banach space c_0.
Thus, for some constant L, there is a map K: X --> c_0 that satisfies the ...
- 4,060
4
votes
5
answers
3k
views
Generalize Fourier transform to other basis than trigonometric function
The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.
In how far can this ...
- 5,327
45
votes
7
answers
16k
views
What is an intuitive view of adjoints? (version 2: functional analysis)
After realising that I don't have an intuitive understanding of adjoint functors, I then realised that I don't have an intuitive understanding of adjoint linear transformations!
Again, I can use 'em, ...
- 26.8k
18
votes
3
answers
2k
views
What are the right categories of finite-dimensional Banach spaces?
This is inspired partly by this question, especially Tom Leinster's answer.
Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who ...
- 11.4k
9
votes
2
answers
1k
views
Explicitly describing extreme points of infinite dimensional convex sets
I am currently trying to apply some results from Choquet theory - i.e., the generalisation of results by Minkowski and Krein-Milman for representing points in a compact, convex set C by probability ...
- 325
6
votes
3
answers
324
views
Inverses in convolution algebras
Let $G$ be a locally compact totally disconnected group, and to make life easy let's suppose its Haar measure is bi-invariant. Let $C_c(G)$ be the space of locally constant complex functions on $G$ ...
- 2,713
11
votes
7
answers
1k
views
What are some interesting ways of making new metrics out of old metrics?
If $d(x,y)$ and $e(x,y)$ are metrics then $d(x,y)+e(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are metrics.
If $d_i(x,y)$ for $i=1,\dots,n$ are metrics then so is $\sqrt{\sum_{i=1}^n{d_i^2(x,y)}}$
Are ...
- 3,613
5
votes
3
answers
230
views
Is the Fell-Doran problem trivial in a topological setting?
The Fell-Doran problem is a problem in functional analysis. It goes as follows: Let $A$ be a complex unital algebra, $X$ a locally convex space, and $L(X)$ the algebra of all continuous endomorphisms ...
- 1,821
3
votes
4
answers
1k
views
Is there a use for a Hilbert space that uses a different norm than the one induced by the inner product?
$l_1$ minimization / compressed sensing comes to mind. Does anyone have any concrete examples? Or is such a construct completely useless?
- 49
6
votes
2
answers
1k
views
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
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
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
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
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
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
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
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\...
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
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
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
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 ...
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
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
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
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
3
votes
1
answer
914
views
Range of a Certain Linear Operator
Consider the following hermitian form on the sobolev space H^1(I), of an interval I:
g(u,v):= \int_I (du/dt dv/dt - \rho(t) u v)dt, where \rho is a nice bounded function on I.
Riesz representation ...
- 31
40
votes
5
answers
10k
views
Is there a natural measures on the space of measurable functions?
Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
- 1,475
21
votes
5
answers
4k
views
Isomorphisms of Banach Spaces
Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a ...
- 629
8
votes
3
answers
698
views
L_p norm balls for 1<p<2 - is it always similar to an L_q norm ball for some q>2?
The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same ...
- 101
2
votes
1
answer
493
views
Convergence of Affine Transformations
Hi,
I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence:
...
- 690
12
votes
4
answers
877
views
Can you describe the image of the exponential map $B(H)\to B(H)$?
James Tener asks at the 20-questions seminar:
The exponential map $\exp:B(H)\to B(H)$ is just defined by its Taylor series. Can you describe its image?
- 1,059