Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
Questions tagged [fa.functional-analysis]
9324 questions
71
votes
2 answers
Barrelled, bornological, ultrabornological, semi-reflexive, ... how are these used?
I'm not a functional analyst (though I like to pretend that I am from time to time) but I use it and I think it's a great subject. But whenever I read about locally convex topological vector spaces, I often get bamboozled by all the different types…
Andrew Stacey
- 26,373
42
votes
7 answers
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, compute 'em, and convolute 'em, but I have no…
Andrew Stacey
- 26,373
26
votes
2 answers
Intuition about L^p spaces
I have read somewhere the following very nice intuition about $L^p(\mathbb{R})$ spaces.
This graphic shows a lot of nice relations:
1) There is no inclusion between $L^p$ and $L^q$
2) $L^p$ is the dual of $L^q$ for $\frac{1}{p} + \frac{1}{q} = 1$,…
cccdi
- 285
25
votes
1 answer
Does there exist a measurable function which is not a.e. "strongly" measurable?
More specifically, letting $I=[0,1]$, do there exist $f,E$ with $E$ a (necessarily nonseparable) Banach space and $f$ a bounded Lebesgue measurable function $I\to E$ such that $f$ is not equal almost everywhere to a pointwise limit of a sequence of…
TaQ
- 3,390
24
votes
2 answers
Dual of the space of Hölder continuous functions?
Let $X=C^{\alpha}(\Omega,\mathbb{R})$ be the space of Hölder continuous functions. What is its dual?
warsaga
- 1,196
23
votes
1 answer
Which Fréchet spaces have a dual that is a Fréchet space?
I've read the claim that Fréchet spaces that are not Banach spaces never have a dual that is a Fréchet space, but have not been able to find a proof of this statement. Is it trivial or does someone have a reference?
Tim van Beek
- 1,544
23
votes
4 answers
Most general definition of differentiation
There are various differentiations/derivatives.
For example,
Exterior derivative $df$ of a smooth function $f:M\to \mathbb{R}$
Differentiation $Tf:TM\to TN$ of a smooth function between manifolds $f:M\to N$
Radon-Nikodym derivative…
Ponta
- 361
23
votes
4 answers
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-meager. So Y must be dense. If Y has the Baire…
Brandon Seward
- 343
21
votes
2 answers
Self-dual normed spaces which are not Hilbert spaces
Are there any examples of non-Hilbert normed spaces which are isomorphic (in the norm sense) to their dual spaces? Or, is there any result in Functional Analysis which says that if a space is self-dual it has to be Hilbert space.
Since, we want…
Uday
- 2,209
21
votes
4 answers
A question on the integral of Hilbert valued functions
This questions stems from an attempt to recast in a form suitable for teaching some standard computations which are usually proved by handwaving, without much care about the details. My hope is that some expert in this area is hanging around and can…
Piero D'Ancona
- 8,775
17
votes
5 answers
A counter example to Hahn-Banach separation theorem of convex sets.
I'm trying to understand the necessity for the assumption in the Hahn-Banach theorem for one of the convex sets to have an interior point. The other way I've seen the theorem stated, one set is closed and the other one compact. My goal is to find a…
Dorian
- 2,601
16
votes
4 answers
Does "taking the dual space" stabilize?
Every book which treats dual spaces of normend spaces states that $(c_0)' = \ell^1$ and $(\ell^1)' = \ell^\infty$ and some also describe $(\ell^\infty)'$.
However, is anything known about higher order duals in general? Does taking the normed dual of…
Dirk
- 12,325
16
votes
1 answer
Unbounded linear operator defined on $l^2$
Let $l^2$ be a Hilbert space of infinite sequences $(z_0, z_1, \cdots)$ with finite $\sum_{i=0}^{\infty} |z_i|^2$.
Are there any simple example of unbounded linear opearator $T: l^2 \to l^2$ with $D(T)=l^2$?
falagar
- 2,761
15
votes
4 answers
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 Theorem is the result that every continuous linear…
Andrew Stacey
- 26,373
15
votes
2 answers
Regularity properties of convolution
Let $f$ be a compactly supported $C^{\alpha}$ function (that is Holder continuous with exponent $\alpha$) and let $g$ be a compactly supported $C^\beta$ function. What can we say about Holder continuity of their convolution
$$
h(x):=\int f(z-x) g…
Oleg
- 921