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Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

10
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2answers
3k 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 $L^{\infty}$ ...
18
votes
3answers
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 ...
2
votes
3answers
2k 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 ...
3
votes
3answers
1k 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}{...
1
vote
1answer
814 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 ...
3
votes
3answers
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
3answers
406 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 ...
23
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2answers
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 ...
21
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4answers
3k 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-...
6
votes
5answers
920 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 ...
10
votes
2answers
746 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 ...
8
votes
5answers
769 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-...
8
votes
1answer
551 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 ...
11
votes
3answers
500 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 ...
20
votes
2answers
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
votes
1answer
1k 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 ...
8
votes
1answer
323 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 ...
2
votes
1answer
890 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 ...
25
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5answers
5k 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 ...
19
votes
5answers
3k 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 ...
7
votes
3answers
671 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 ...
1
vote
1answer
281 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: ...
10
votes
4answers
713 views

Can you describe the image of the exponential map B(H)->B(H).

James Tener asks at the 20-questions seminar: The exponential map exp:B(H)->B(H) is just defined by its Taylor series. Can you describe its image?