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Questions tagged [fa.functional-analysis]

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

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
405 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
votes
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
votes
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
910 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
745 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
763 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
550 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
498 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
321 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
888 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
votes
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
668 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
709 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?