Questions tagged [fa.functional-analysis]

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

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18 answers
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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 ...
92 votes
1 answer
10k 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 ...
algori's user avatar
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79 votes
4 answers
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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 ...
Jonas Meyer's user avatar
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78 votes
3 answers
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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 $\...
Bill Johnson's user avatar
76 votes
0 answers
4k 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\...
Ady's user avatar
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71 votes
16 answers
20k 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 ...
71 votes
2 answers
6k 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, ...
Andrew Stacey's user avatar
67 votes
3 answers
11k views

Nonconvexity and discretization

Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem. Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
Peter Scholze's user avatar
65 votes
9 answers
11k views

Polish spaces in probability

Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed. Question: What can go wrong when doing probability on non-Polish spaces?
Thanh's user avatar
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65 votes
14 answers
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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 ...
jon's user avatar
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62 votes
5 answers
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Jean Bourgain's relatively lesser known significant contributions

Jean Bourgain passed away on December 22, 2018. A great mathematician is no longer with us. Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions,...
61 votes
7 answers
9k 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 ...
teil's user avatar
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58 votes
9 answers
9k 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 ...
shuhalo's user avatar
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57 votes
23 answers
101k views

A good book of functional analysis [closed]

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)....
55 votes
7 answers
27k 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 ...
51 votes
2 answers
4k 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 ...
Nathaniel Johnston's user avatar
49 votes
7 answers
16k 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 ...
Orbicular's user avatar
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48 votes
6 answers
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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 "...
47 votes
3 answers
12k views

Pullback measures

Why do all measure theory textbooks present the concept of push-forward measure, but never the concept of pull-back measure? Doesn't the latter exist? It's true that the naive treatment of such a ...
Alex M.'s user avatar
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47 votes
6 answers
5k 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 ...
Vidit Nanda's user avatar
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47 votes
0 answers
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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 $l^2$...
Nik Weaver's user avatar
45 votes
1 answer
4k 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 ...
Theo Buehler's user avatar
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44 votes
7 answers
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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 ...
Andrew Stacey's user avatar
44 votes
1 answer
2k views

Existence and uniqueness of Haar measure on compacta; a cohomological approach

I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group. I think the best way of introducing the idea I am pursuing is ...
Ronald J. Zallman's user avatar
43 votes
10 answers
45k views

Is square of Delta function defined somewhere?

I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere. In the beginning, this question might look strange. But by restricting the space of the test functions, ...
43 votes
3 answers
8k views

Why the name 'separable' space?

It is well known that a separable space is a topological space that has a countable dense subset. I am wondering how is this related to the name 'separable'? Any intuition where the name come from?
minimax's user avatar
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42 votes
1 answer
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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?
Lost's user avatar
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41 votes
7 answers
14k 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, ...
Andrew Stacey's user avatar
41 votes
5 answers
5k 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(...
john mangual's user avatar
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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 ...
Kenny Easwaran's user avatar
39 votes
3 answers
13k views

Is the Invariant Subspace Problem interesting?

There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the Invariant Subspace Problem, he says (paraphrasing) "...this question is still open. It is also an ...
William DeMeo's user avatar
39 votes
4 answers
15k 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 ...
Bill Johnson's user avatar
37 votes
2 answers
12k views

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

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] Louis ...
Zen Harper's user avatar
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37 votes
2 answers
2k 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 ...
fedja's user avatar
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36 votes
4 answers
4k 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{\...
Thomas Rot's user avatar
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35 votes
4 answers
5k 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 ...
Daan Michiels's user avatar
35 votes
2 answers
1k 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 ...
David E Speyer's user avatar
35 votes
2 answers
8k 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 ...
André Henriques's user avatar
34 votes
8 answers
8k views

When is a Banach space a Hilbert space?

Let $\mathcal{X}$ be a real or complex 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. ...
Teiko Heinosaari's user avatar
34 votes
3 answers
3k 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 ...
Dmitri Pavlov's user avatar
34 votes
2 answers
907 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 ...
Elle Najt's user avatar
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34 votes
1 answer
3k 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 $ab$ ...
Bill Johnson's user avatar
33 votes
3 answers
3k views

Reference request for translating from Top to C*-alg

Some recent questions on MO (for example, Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?) have been about Gelfand duality — namely, that the categories of ...
Matthew Daws's user avatar
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33 votes
4 answers
2k views

Hahn-Banach theorem with convex majorant

At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and $f:L\...
Jochen Wengenroth's user avatar
33 votes
1 answer
4k views

Theme of Isbell duality

Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$....
Martin Brandenburg's user avatar
33 votes
0 answers
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-...
Andreas Thom's user avatar
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32 votes
3 answers
2k 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 ...
FusRoDah's user avatar
  • 3,680
32 votes
2 answers
4k 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 ...
diverietti's user avatar
  • 7,852
32 votes
1 answer
2k 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^...
André Henriques's user avatar
32 votes
1 answer
2k 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 ...
Sergei Akbarov's user avatar

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