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Questions tagged [fa.functional-analysis]

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

108
votes
17answers
11k views

Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
76
votes
1answer
7k views

The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly. Arguably, they are the most important such tool. Briefly, the question I'd like to ...
71
votes
3answers
6k views

Norms of Commutators

If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
64
votes
0answers
3k views

2, 3, and 4 (a possible fixed point result ?)

The question below is related to the classical Browder-Goehde-Kirk fixed point theorem. Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$ be a mapping such that $\Vert Tx-Ty\...
63
votes
4answers
5k views

Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I: The theory of commutative ...
60
votes
2answers
4k views

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, ...
52
votes
15answers
12k views

Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.] I've read looked at the examples in most ...
51
votes
13answers
3k 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 ...
48
votes
7answers
6k views

Why is the Hahn-Banach theorem so important?

Every time I hear it mentioned it is praised in the highest possible terms, and I remember one of my old lecturers saying that it is one of the 3 most important theorems in analysis. Yet the only ...
47
votes
9answers
6k views

Motivation for and history of pseudo-differential operators

Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...
45
votes
6answers
4k views

Can we actually find any fixed points with Brouwer's theorem?

Background At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...
44
votes
2answers
2k views

A strengthening of the Cauchy-Schwarz inequality

Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the ...
43
votes
0answers
2k views

Set-theoretic reformulation of the invariant subspace problem

The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...
41
votes
5answers
5k views

Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?

Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway... Consider this problem. I've been trying to find a formula to expand the "regular iteration" of "...
39
votes
8answers
5k views

Polish spaces in probability

Probabilist are very often working with Polish spaces, though this is not always very clear where this assumption is needed. question: what can go wrong when doing probability on non-Polish spaces ?
38
votes
0answers
3k views

Example of a compact set that isn't the spectrum of an operator

This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity: Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty ...
37
votes
7answers
5k 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 ...
37
votes
7answers
12k views

Way to memorize relations between the Sobolev spaces?

Consider the Sobolev spaces $W^{k,p}(\Omega)$ with a bounded domain $\Omega$ in n-dimensional Euclidean space. When facing the different embedding theorems for the first time, one can certainly feel ...
36
votes
21answers
53k views

A good book of functional analysis

I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the basics)....
36
votes
2answers
1k views

Moving one family of commuting self-adjoint operators to another without losing commutativity on the way

This is actually not a question of mine, so I'll be short on motivation and say nothing beyond that if this were true, a few fancy harmonic analysis techniques that a colleague of mine used in proving ...
35
votes
9answers
27k views

Is square of Delta function defined somewhere?

Hello, every one. I am wondering whether any one knows that whether the square of Dirac Delta function is defined some where? In the beginning, this question might look strange. But by restricting ...
35
votes
6answers
14k views

Learning roadmap for harmonic analysis

In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
34
votes
1answer
9k views

What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

Hi, I know this is a dangerous topic which could attract many cranks and nutters, but: According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] ...
34
votes
1answer
2k views

tr(ab)=tr(ba), part 2.

This is a Banach space version of Andre Henriques' question Trace Question for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ ...
33
votes
2answers
5k views

tr(ab) = tr(ba)?

It is well known that given two Hilbert-Schmidt operators $a$ and $b$ on a Hilbert space $H$, their product is trace class and $tr(ab)=tr(ba)$. A similar result holds for $a$ bounded and $b$ trace ...
32
votes
4answers
3k views

“Entropy” proof of Brunn-Minkowski Inequality?

I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality. The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then $$ m(...
32
votes
4answers
3k views

Which differential equations allow for a variational formulation?

Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation $$ \frac{d}{dt}\frac{\...
32
votes
1answer
1k views

For which maps $S^1\to S^1$ is the winding number defined?

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number: • Continuous maps: Using the unique path lifting property of the universal covering map $\mathbb R\to S^...
32
votes
2answers
667 views

If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$?

I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...
32
votes
0answers
1k views

Subalgebras of von Neumann algebras

In the late 70s, Cuntz and Behncke had a paper H. Behncke and J. Cuntz, Local Completeness of Operator Algebras, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (Jan., 1977), pp. 95-...
31
votes
2answers
803 views

Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?

Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
31
votes
1answer
3k views

Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic?

Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?
31
votes
2answers
3k views

Are there non-reflexive vector spaces isomorphic to their bi-dual?

Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{**}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism. Are ...
30
votes
3answers
1k views

Why are there so many fractional derivatives?

I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator. I started with the book The Fractional Calculus ...
29
votes
8answers
6k views

When is a Banach space a Hilbert space?

Let $\mathcal{X}$ be a Banach space. It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds. Question: Are there ...
29
votes
4answers
9k views

Product of Borel sigma algebras

If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I ...
29
votes
3answers
2k views

Can we recover a von Neumann algebra from its predual?

By definition, a von Neumann algebra is a C*‑algebra A that admits a predual, i.e., a Banach space Z such that Z* is isomorphic to the underlying Banach space of A. (We require that isomorphisms in ...
28
votes
2answers
1k views

Can an operator have Exp(z) as its characteristic “polynomial”?

Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator. Define $$ f_T(z) = \sum_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$ the ...
28
votes
0answers
998 views

When are two C*-algebras isomorphic as Banach spaces?

We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its ...
27
votes
11answers
17k views

A book for problems in Functional Analysis

I want to know if there's any book that categorizes problems by subjects of Functional Analysis. I'm studying Functional Analysis now a days and I really need to solve some problems in order to ...
27
votes
2answers
2k views

Is a normed space which is homeomorphic to a Banach space complete?

I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$. Does this imply that $(E,||\cdot||)$ is also a Banach space? I think I read something ...
27
votes
3answers
2k views

Surjectivity of operators on $\ell^\infty$

Can anyone give me an example of an bounded and linear operator $T:\ell^\infty\to \ell^\infty$ (the space of bounded sequences with the usual sup-norm), such that T has dense range, but is not ...
27
votes
1answer
1k views

Stone-Weierstrass theorem for holomorphic functions?

The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called Naсhbin's theorem: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth ...
27
votes
2answers
2k views

Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function. Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e. $\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all $x\in\...
27
votes
1answer
1k views

Szőkefalvi-Nagy's unitarizability theorem in the Calkin algebra?

Here's a research problem, which I think interesting. Suppose that $t$ is an invertible element in the Calkin algebra $\mathcal{Q} = \mathcal{B}(\ell_2)/\mathcal{K}(\ell_2)$ which satisfies $\sup_{n \...
27
votes
0answers
976 views

Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $||x|| = 1$ and $||Ax|| = ||A||$. The ...
26
votes
15answers
4k views

Important results that use infinite-dimensional manifolds?

Are Banach manifolds (or other types of infinite-dimensional manifolds) just curiosities, or have they been utilized to prove some interesting/important results? Where do they turn up? Important ...
26
votes
7answers
2k views

Intuition for failure of Implicit Function theorem on Frechet Manifolds

When dealing with moduli spaces of, say connections or metrics, I am using the notions of Frechet spaces/manifolds/groups. I have become familiar with Banach manifolds (I think), but Frechet manifolds ...
26
votes
9answers
3k views

Applications of algebra to analysis

EDIT: I would like to make a list of modern applications of algebra in analysis. By "modern" I will mean developments since the beginning of the 20th century. It is well known that classical linear ...
26
votes
4answers
2k views

How are infinite-dimensional manifolds most commonly treated?

I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I am still looking for ...