All Questions
10,826 questions
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 $\...
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
$$X = \left\{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{...
18
votes
1
answer
1k
views
Are there non-reflexive abelian topological groups isomorphic to their second dual?
I posted the following question in a comment at
Are there non-reflexive vector spaces isomorphic to their bi-dual? and it got one upvote, but it didn't get an answer, so I'll post it as an ...
17
votes
5
answers
7k
views
A counter example to Hahn-Banach separation theorem of convex sets.
I'm trying to understand the necessity for the assumption in the Hahn-Banach theorem for one of the convex sets to have an interior point. The other way I've seen the theorem stated, one set is closed ...
17
votes
4
answers
959
views
What is the minimum of this quantity on $S^{n-2}\times S^{n-2}$?
My question is to find the minimum of the following expression:
$$A(x,y) = \sum_{1\leq i<j\leq n} |x_i-x_j|\ |y_i-y_j|,$$
over the set of pairs of real vectors $x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)$ ...
17
votes
3
answers
2k
views
Is every Schwartz function the product of two Schwartz functions?
A Schwartz function on $\mathbb R^d$ is a $C^\infty$ function, such that all differentials of order $k \ge 0$ decay faster than any polynomial. They include the class $C^\infty_c(\mathbb R^d)$ of ...
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
17
votes
2
answers
834
views
When is $\sum_{n\in\mathbb Z} f(x+n)$ constant?
A recently asked question (linked here) deals with the remarkable identity
$$ \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 ...
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.
...
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 ...
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 ...
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 $\...
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 $\...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
...
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 ...
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 $\...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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. ...
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
\...
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 ...
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\...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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
...
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 ...
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} $$
where I'm keeping things simple by ...
16
votes
6
answers
2k
views
Finding closed subspaces whose sum isn't closed
Let $V_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum ...
16
votes
1
answer
3k
views
What is the intuition behind Almgren's frequency function?
It is by now well-known that for a harmonic function $u : B_1^n(0) \to \mathbb{R}$, the ratio
$$
N(r) := \frac{r\int_{B_r(0)}|\nabla u|^2}{\int_{\partial B_r(0)} u^2}
$$
is a non-decreasing function ...
16
votes
1
answer
969
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
16
votes
2
answers
682
views
Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?
Let $X$ be a Banach space. Consider the map
$$
\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\...
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{...
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 ...