Questions tagged [fa.functional-analysis]

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

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40
votes
7answers
6k 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 ...
32
votes
4answers
11k 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 ...
20
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5answers
2k views

Nice applications for Schwartz distributions

I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are: Some multilinear algebra including the Kernel Theorem and ...
12
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2answers
1k views

Question about functional derivatives

This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a ...
29
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5answers
7k 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 ...
20
votes
2answers
7k views

Fourier transform of the unit sphere

The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula $$ \int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
21
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6answers
5k views

prime ideals in C([0,1])

It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa. So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, ...
9
votes
2answers
629 views

Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
10
votes
1answer
887 views

Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$

Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...
3
votes
1answer
377 views

Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$? Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
22
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2answers
5k views

Compact embeddings of Sobolev spaces: a counterexample showing the Rellich-Kondrachov theorem is sharp

Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...
26
votes
1answer
2k views

Criteria for boundedness of power series

Consider a power series $\sum_{n=0}^{\infty} a_n x^n$ that is convergent for all real x, thus defining a function $f: \mathbb{R} \to \mathbb{R}$. Can one give necessary and sufficient criteria the ...
17
votes
5answers
4k views

Hahn-Banach without Choice

The standard proof of the Hahn-Banach theorem makes use of Zorn's lemma. I hear that, however, Hahn-Banach is strictly weaker than Choice. A quick search leads to many sources stating that Hahn-Banach ...
12
votes
5answers
2k views

Measure theory treatment geared toward the Riesz representation theorem

I'm looking for recommendations for books (or lecture notes) that develop measure theory in sufficient detail to state and prove the Riesz representation theorem (which is the characterization of the ...
21
votes
1answer
2k views

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 ...
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-...
20
votes
6answers
6k views

Almost orthogonal vectors

This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
11
votes
4answers
1k views

Norm continuous infinite dimenisonal representation of a Lie group

Given a Lie group G and an infinite dimensional Hilbert space $\mathcal{H}$. In the literature I have only encountered the two following notions of a representation $\pi$ of G on $\mathcal{H}$ : 1) $\...
19
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6answers
2k views

Can distribution theory be developed Riemann-free?

I imagine most people who frequent MO have been indoctrinated into the point of view that the Riemann integral can be safely discarded once one has taken the time to develop the Lebesgue integral. ...
14
votes
3answers
2k views

Universal $C^*$-algebra with generators and relations

We say that the $C^*$-algebra $A$ generated by $a_1,...,a_n$ is universal subject to relations $R_1,...,R_m$ if for every $C^*$-algebra $B$ with elements $b_1,...,b_n$ satisfying relations $R_1,...,...
8
votes
1answer
436 views

Can $L^1_{loc}$ be represented as colimit?

Let $L^1_{loc}$ denote the set of all functions from $\mathbb{R}$ to itself which are locally integrable. For every infinite compact subset $K\subseteq \mathbb{R}$, let $L^1_{m_K}$ denote the space ...
6
votes
2answers
420 views

Harmonic oscillator discrete spectrum

Let us act intentionally stupid and assume we do not know that we can solve for the spectrum of the harmonic oscillator $$-\frac{d^2}{dx^2}+x^2$$ explicitly. Is there an abstract argument why the ...
5
votes
1answer
499 views

Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$

The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to $$ -\Delta u=f\hspace{3cm}(1)? $$ I'm of ...
5
votes
0answers
151 views

Heuristic and graphic representation of BV functions and their singularities

This question is about some heuristics and graphs of BV functions. In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are the Heaviside function, whose ...
2
votes
2answers
644 views

Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$ Is there any information ...
13
votes
2answers
1k views

Calkin Algebra and the embedding

Let $H$ be a separable, infinite dimensional Hilbert Space and $Calk(H):=B(H)/K(H)$ denotes the Calkin algebra. There is obvious surjection $\pi: B(H) \to Calk(H)$ but I'm interested in somehow ...
8
votes
3answers
1k views

Generalizations and relative applications of Fekete's subadditive lemma

Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...
3
votes
1answer
221 views

Hausdorff dimension of the graph of a BV function

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function. Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it? Update. In an answer to this post, it ...
8
votes
1answer
531 views

A non-hyperfinite type III factor from an action of the free group on the circle

We define below a von Neumann algebra $\mathcal{M}$ from an action of the free group on the circle, and we prove that $\mathcal{M}$ is a non-hyperfinite type ${\rm III}$ factor. Question : Is $\...
6
votes
1answer
362 views

Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$

How is the proof that $$[L^2(0,T;X)]' = L^2(0,T;X')$$ looking like, where $X$ is a Hilbert space? I am asking for the proof that the dual space of $L^2(0,T;X)$ is the space $L^2(0,T;X^*)$. Is the ...
5
votes
2answers
852 views

Is the space of signed finite measures on a compact set $M([0,1])$ a sequential space?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\...
4
votes
1answer
191 views

Universal decay rate of the Fisher information along the heat flow

I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation $$ \partial_t u=\Delta u $$ ...
2
votes
2answers
246 views

Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative? More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of a function $$...
34
votes
4answers
4k 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(...
33
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{\...
46
votes
2answers
3k 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 ...
55
votes
5answers
6k views

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,...
46
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 ...
37
votes
9answers
33k views

Is square of Delta function defined somewhere?

everyone. 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 ...
23
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3answers
1k views

Universality of zeta- and L-functions

Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
47
votes
6answers
6k 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 "...
34
votes
1answer
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 $ab$ ...
15
votes
2answers
1k views

States in C*-algebras and their origin in physics?

in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$ Now, of course there is also in classical physics and quantum ...
17
votes
2answers
3k views

Ideals of the ring of smooth functions

The ring $C^\infty(M)$ of smooth functions on a smooth manifold $M$ is a topological ring with respect to the Whitney topology and the usual ring operations. Is it possible to describe, maybe under ...
33
votes
2answers
7k 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 ...
33
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?
27
votes
7answers
9k views

Regular borel measures on metric spaces

When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the ...
25
votes
2answers
2k views

$f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$

Let $f$ be a real function with domain R. If $f^2$ and $f^3$ are both infinitely differentiable on R, how to prove $f$ is infinitely differentiable on R? I have been thinking about this problem for a ...
12
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3answers
1k views

Disintegrations are measurable measures - when are they continuous?

This is a sequel to another question I have asked. The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
16
votes
1answer
1k views

Bases for spaces of smooth functions

Let $S$ denote the space of rapidly decreasing sequences, which means sequences $a=(a_k)_{k=1}^\infty$ such that the numbers $p_d(a)=\sup\{k^d|a_k| : 1\leq k<\infty\}$ are finite for all $d\in\...

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