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18 votes
1 answer
996 views

Existance of certain almost invariant functions related to amenability and piece-wise transformations

We would like very much to know the answer to the following question: Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
18 votes
2 answers
1k views

Complex structure on $L^2(\mathbb R)$ generalizing the Hilbert transform

The Hilbert transform on the real Hilbert space $L^2(\mathbb R)$ is the singular integral operator $$ \mathcal H(f)(x) := \frac{1}{\pi} \int_{-\infty}^\infty \frac{1}{x-y} f(y) dy. $$ It satisfies $\...
André Henriques's user avatar
18 votes
1 answer
2k views

Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain

Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space $$X = \left\{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{...
JakeA's user avatar
  • 201
18 votes
1 answer
1k views

Are there non-reflexive abelian topological groups isomorphic to their second dual?

I posted the following question in a comment at Are there non-reflexive vector spaces isomorphic to their bi-dual? and it got one upvote, but it didn't get an answer, so I'll post it as an ...
KConrad's user avatar
  • 50.6k
17 votes
5 answers
7k views

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 ...
Dorian's user avatar
  • 2,641
17 votes
4 answers
959 views

What is the minimum of this quantity on $S^{n-2}\times S^{n-2}$?

My question is to find the minimum of the following expression: $$A(x,y) = \sum_{1\leq i<j\leq n} |x_i-x_j|\ |y_i-y_j|,$$ over the set of pairs of real vectors $x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)$ ...
Mostafa - Free Palestine's user avatar
17 votes
3 answers
2k views

Is every Schwartz function the product of two Schwartz functions?

A Schwartz function on $\mathbb R^d$ is a $C^\infty$ function, such that all differentials of order $k \ge 0$ decay faster than any polynomial. They include the class $C^\infty_c(\mathbb R^d)$ of ...
Paul Pfeiffer's user avatar
17 votes
2 answers
5k views

Positive-Definite Functions and Fourier Transforms

Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite. ...
Alex R.'s user avatar
  • 4,952
17 votes
2 answers
834 views

When is $\sum_{n\in\mathbb Z} f(x+n)$ constant?

A recently asked question (linked here) deals with the remarkable identity $$ \sum_{n\in\mathbb Z} \mathrm{sinc}(n+x)= \pi,\quad x\in\mathbb R, $$ where $\mathrm{sinc}(x)=\sin(x)/x$. It is easy ...
W-t-P's user avatar
  • 550
17 votes
4 answers
2k views

What are the major differences between real and complex Banach space?

Most theorems under real Banach space settings have their twin brothers for complex ones, say, the Hahn-Banach theorem. However, some theorems are not valid in complex Banach spaces, and vice versa. ...
Ice sea's user avatar
  • 407
17 votes
3 answers
3k views

Which sigma-ideals in a sigma-algebra are ideals of null sets?

My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space ...
Super-Measurable Analyst's user avatar
17 votes
3 answers
3k views

Why is multiplication on the space of smooth functions with compact support continuous?

I asked the question Why is multiplication on the space of smooth functions with compact support continuous? on M.SE sometime ago but I didn't receive a satisfactory answer. I was reading this ...
Hugo's user avatar
  • 394
17 votes
1 answer
986 views

Uncountably many subsets of the natural numbers with certain natural density condition

Are there uncountably many $A_\alpha $ of subsets of $\mathbb{N}$ with the following two properties: Each $A_\alpha$ has positive upper natural density $A_\alpha \cap A_\beta$ is a finite set for $\...
Ali Taghavi's user avatar
17 votes
3 answers
975 views

Evaluating the sum $f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n$ and estimating bounds

For real variable $x$, the function \begin{equation} f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n \end{equation} clearly has infinite radius of convergence and defines a $C^\infty$ function on $\...
Isaac's user avatar
  • 3,477
17 votes
1 answer
861 views

Extreme points of convex compact sets

Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an ...
Fedor Petrov's user avatar
17 votes
1 answer
1k views

How many values determine a norm?

It is well known that for a bilinear form over an n-dimensional vector space, $n^2$ values (on all pairs of basis-vectors) determine it uniquely. How many values do we need to specify in order to ...
Asaf Shachar's user avatar
  • 6,741
17 votes
1 answer
2k views

Are "most" operators on an infinite-dimensional complex Banach space "diagonalizable"?

This is true for finite-dimensional spaces: the diagonal operators on a finite dimensional complex vector space form contain a dense open set and the nondiagonalizable operators have measure 0. To be ...
Tim Campion's user avatar
  • 63.9k
17 votes
4 answers
2k views

Where do the real analytic Eisenstein series live?

In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a ...
Eren Mehmet Kiral's user avatar
17 votes
2 answers
2k views

Realisation of the noncommutative torus as a universal $ C^{*} $-algebra

One of the most basic examples in noncommutative geometry is the so-called noncommutative torus, denoted here by $ \mathbb{T}_{\theta} $. As far as I know, there are several equivalent constructions ...
truebaran's user avatar
  • 9,330
17 votes
3 answers
905 views

Existence of translation-invariant basis on $C_c(\mathbb R)$

Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
Nick S's user avatar
  • 2,071
17 votes
4 answers
2k views

Banach-Mazur applied to a Hilbert space

The Banach-Mazur theorem says that every separable Banach space is isometric to a subspace of $C^0([0;1],R)$, the space of continuous real valued functions on the interval $[0;1]$, with the sup norm. ...
Laurent Berger's user avatar
17 votes
4 answers
4k views

How much does the absolute value of an operator behave like an absolute value?

Recall that the absolute value of a bounded operator $T$ on a Hilbert space $H$ is the unique positive operator $|T|$ such that $$\||T|x\|=\|Tx\|$$ for all $x\in H$. It can be defined using the ...
Iian Smythe's user avatar
  • 3,115
17 votes
5 answers
3k views

Conditional probabilities are measurable functions - when are they continuous?

Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
Tom LaGatta's user avatar
  • 8,512
17 votes
1 answer
2k views

Which Fréchet manifolds have a smooth partition of unity?

A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is: Which Fréchet manifolds have a smooth partition of unity? How is the ...
Konrad Waldorf's user avatar
17 votes
1 answer
569 views

Does a completely metrizable space admit a compatible metric where all intersections of nested closed balls are non-empty?

(cross-posted from this math.SE question) It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with respect to ...
Cla's user avatar
  • 775
17 votes
1 answer
807 views

Operator Valued Kadison--Singer Problem

The Paving Conjecture, which is equivalent to the famous Kadison--Singer Problem, was spectacularly settled in the affirmative by Marcus--Spielman--Srivastava (arxiv:1306.3969). Let $E$ denote the ...
Narutaka OZAWA's user avatar
17 votes
1 answer
912 views

$(1+\epsilon)$-injective Banach spaces, complex scalars

It is well known that a real Banach space which is $(1+\epsilon)$-injective for every $\epsilon >0$ is already 1-injective (Lindenstrauss Memoirs AMS, 1964, download here). Using common ...
Fred Dashiell's user avatar
17 votes
1 answer
759 views

Classification of non-Hausdorff topological vector spaces

It is well-known that up to topological isomorphism there is exactly one Hausdorff topological vector space (say, over $\mathbb{C}$) of a given dimension $n$, namely $\mathbb{C}^n$ with the euclidean ...
HeinrichD's user avatar
  • 5,482
17 votes
2 answers
953 views

Convexity of spectral radius of Markov operators, Random walks on non-amenable groups

Let $P_1,P_2$ denote stochastic transition matrices on a countable set $I$. Consider $P_1,P_2$ as operators on $\ell^2(I)$ given by multiplication. Question Under which conditions can we show that ...
Mika's user avatar
  • 171
17 votes
2 answers
4k views

Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
Rajesh D's user avatar
  • 698
17 votes
3 answers
770 views

Does a spectral gap lift to covering spaces?

Let $M$ be a complete Riemannian manifold. Denote $\Delta_M\ge0$ the unique self-adjoint extension of the Laplace-Beltrami operator in $L^2(M)$ and $\sigma(\Delta_M)\subset [0,\infty)$ its spectrum. ...
Jan Bohr's user avatar
  • 779
17 votes
1 answer
1k views

Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates \...
Tobias Diez's user avatar
  • 5,824
17 votes
0 answers
677 views

Are dualizable topological vector spaces finite-dimensional?

Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product. Every finite-dimensional ...
Dmitri Pavlov's user avatar
17 votes
0 answers
488 views

Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to (0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set $\{x_i\...
Mikhail Ostrovskii's user avatar
16 votes
3 answers
2k views

Integration of a function over 7-sphere

Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$. The problem is finding or approximating the ...
Hrushikesh Pawar's user avatar
16 votes
4 answers
2k views

Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure?

This question arose a few years back when I was an assistant teacher on a course of basic (Lebesgue) measure theory, but I didn't find an answer or anyone able to solve the problem. The setting of the ...
Rami Luisto's user avatar
16 votes
2 answers
1k views

How to generalize the various vector calculus theorems to distributions?

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
YuerWu's user avatar
  • 415
16 votes
1 answer
3k views

Where did the term "additive energy" originate?

A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the additive energy of $A$ is defined to be the ...
Mark Lewko's user avatar
16 votes
4 answers
7k views

Good book for measure theory and functional analysis

I have taken advanced courses both in measure theory and also in functional analysis (Banach and Hilbert spaces, spectral theory of bounded and unbounded operators, etc.) The connections between the ...
Saeid Haghighatshoar's user avatar
16 votes
3 answers
950 views

What are the 'wonderful consequences' following from the existence of a minimal dense subspace?

In Peddechio & Tholens Categorical Foundations they quote PT Johnstone in their chapter on Frames & Locales: ...the single most important fact which distinguishes locales from spaces: the ...
Mozibur Ullah's user avatar
16 votes
3 answers
708 views

An inequality for two independent identically distributed random vectors in a normed space

Suppose that $X$ and $Y$ are independent identically distributed random vectors in a separable Banach space $B$. Does it always follow that $E\|X-Y\|\le E\|X+Y\|$? Some background information on ...
Iosif Pinelis's user avatar
16 votes
2 answers
996 views

Perturbation of unbounded self-adjoint operators

In the paper "A CRITERION FOR THE NORMALITY OF UNBOUNDED OPERATORS AND APPLICATIONS TO SELF-ADJOINTNESS" by M. H MORTAD (http://arxiv.org/pdf/1301.0241.pdf), the author states the following theorem ...
m.gn's user avatar
  • 163
16 votes
3 answers
1k views

A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$. Motivation: A lot! For example, in game theory $S$ can be a set of ...
Bogdan's user avatar
  • 161
16 votes
3 answers
852 views

Quantum Hamiltonian for an Inverse Cube Force Law

If you have a nonrelativistic quantum particle in $\mathbb{R}^3$ in an attractive inverse cube force, its Hamiltonian is $$ H = -\nabla^2 - \frac{c}{r^2} $$ where I'm keeping things simple by ...
John Baez's user avatar
  • 22.3k
16 votes
6 answers
2k views

Finding closed subspaces whose sum isn't closed

Let $V_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum ...
Nik Weaver's user avatar
  • 42.8k
16 votes
1 answer
3k views

What is the intuition behind Almgren's frequency function?

It is by now well-known that for a harmonic function $u : B_1^n(0) \to \mathbb{R}$, the ratio $$ N(r) := \frac{r\int_{B_r(0)}|\nabla u|^2}{\int_{\partial B_r(0)} u^2} $$ is a non-decreasing function ...
SBK's user avatar
  • 1,179
16 votes
1 answer
969 views

Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis

While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
Tobias Diez's user avatar
  • 5,824
16 votes
2 answers
682 views

Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?

Let $X$ be a Banach space. Consider the map $$ \alpha\colon X\hat{\otimes} X^* \to B(X)^*, $$ defined one simple tensors as $$ \alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, a\...
Hannes Thiel's user avatar
  • 3,497
16 votes
3 answers
5k views

Integration of differential forms using measure theory?

Setup: Let $(M,g)$ be a (possibly non-compact) Riemannian manifold with volume density $d_gV$. Then one may think of $(M,g)$ as a measure space $(\Omega,\mathcal{A},\mu)$, where $\Omega:=M$, $\mathcal{...
Meneldur's user avatar
  • 408
16 votes
2 answers
4k views

Usefulness of Frechet versus Gateaux differentiability or something in between.

If you have a function $V: L \rightarrow \mathbb{R}$, where $L$ is an infinite dimensional topological vector space, there are multiple notions of differentiability. For $x,u \in L$, $V$ is Gateaux ...
weakstar's user avatar
  • 943

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