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19 votes
2 answers
2k views

Can we take a supremum over all Hilbert spaces?

In my paper On the optimal error bound for the first step in the method of cyclic alternating projections, I defined functions $f_n:[0,1]\to\mathbb{R}$, $n\geqslant 2$, by $$ f_n(c)=\sup\{\|P_n\dotsm ...
Ivan Feshchenko's user avatar
19 votes
5 answers
16k views

What does "kernel" mean in integral kernel?

In functional analysis, there is the term "integral kernel". Examples are Possion kernel, Dirichlet kernel etc. In algebra, the term kernel of a homomorphism refers to the inverse image of the zero ...
user avatar
19 votes
1 answer
5k views

A Fourier-analytic inequality used by Jean Bourgain

I am currently reading Jean Bourgain's 1986 paper A Szemerédi type theorem for sets of positive density in $R^k$ and would appreciate some help in understanding a Fourier-analytic estimate used in ...
Ian Morris's user avatar
  • 6,206
19 votes
7 answers
2k views

Generalizations of "standard" calculus

We have the usual analogy between infinitesimal calculus (integrals and derivatives) and finite calculus (sums and forward differences), and also the generalization of infinitesimal calculus to ...
Zev Chonoles's user avatar
  • 6,792
19 votes
3 answers
1k views

What standard Banach space is isomorphic to the completion of this different normed structure on $\ell^1$?

A colleague asked me the following question: "What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?" This ...
Ali Taghavi's user avatar
19 votes
4 answers
5k views

Explicit extension of Lipschitz function (Kirszbraun theorem)

Kirszbraun theorem states that if $U$ is a subset of some Hilbert space $H_1$, and $H_2$ is another Hilbert space, and $f : U \to H_2$ is a Lipschitz-continuous map, then $f$ can be extended to a ...
gondolier's user avatar
  • 1,839
19 votes
1 answer
773 views

Are algebraically isomorphic $C^*$-algebras $*$-isomorphic?

If A and B are C^*-algebras that are algebraically isomorphic to each other, does this imply that they are *-isomorphic to each other?
Doc Matrix's user avatar
19 votes
1 answer
5k views

Intuition for the Hardy space $H^1$ on $R^n$

the standard intuition for Lebesgue spaces $L^p(\mathbb R^n)$ for $p \in [1,\infty]$ are measurable functions with certain decay properties at infinity or at the singularities. In particular, a ...
shuhalo's user avatar
  • 5,327
19 votes
2 answers
5k views

Is there an infinite-dimensional Banach space with a compact unit ball?

A popular pair of exercises in first courses on functional analysis prove the following theorem: The unit ball of a Banach space $X$ is compact if and only if $X$ is finite-dimensional. My ...
Mark Meckes's user avatar
  • 11.4k
19 votes
3 answers
1k views

Is there "Schur-Weyl duality" for infinite dimensional unitary group?

To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
Michał Oszmaniec's user avatar
19 votes
3 answers
711 views

Almost isometric linear maps

Say that a linear map $\varphi : B(\mathcal H) \rightarrow B(\mathcal H)$ is $\epsilon$-almost isometric if $$ 1 - \epsilon \leq \lVert\varphi(a)\rVert \leq 1+\epsilon, \quad \forall a\in B(\mathcal H)...
Chris Ramsey's user avatar
  • 3,984
19 votes
1 answer
3k views

Infinite convex combinations in a Banach space

Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds: For any sequence $(x_k)_{k\ge0}$ in $C$, and for any sequence of non-negative real numbers $(\...
Pietro Majer's user avatar
  • 60.5k
19 votes
0 answers
552 views

Talagrand's "Creating convexity" conjecture

We say a subset $A$ of $\mathbb{R}^N$ is balanced if \begin{equation} x \in A, \lambda \in [-1,1] \implies \lambda x \in A. \end{equation} Given a subset $A$ of $\mathbb{R}^N$, we write \begin{...
Samuel Johnston's user avatar
18 votes
6 answers
4k views

What is the best place to learn about the mathematical foundations of quantum mechanics?

I'm looking for good references to learn about the mathematical foundations of quantum mechanics. By mathematical foundations, I do not mean rigorous quantum mechanics in general but the axioms behind ...
MathMath's user avatar
  • 1,305
18 votes
3 answers
2k views

Research topics in distribution theory

The theory of distributions is very interesting, and I have noticed that it has many applications especially with regard to PDEs. But what are the research topics in this theory? also in terms of ...
Andrew's user avatar
  • 589
18 votes
4 answers
2k views

Does "taking the dual space" stabilize?

Every book which treats dual spaces of normend spaces states that $(c_0)' = \ell^1$ and $(\ell^1)' = \ell^\infty$ and some also describe $(\ell^\infty)'$. However, is anything known about higher ...
Dirk's user avatar
  • 12.7k
18 votes
3 answers
2k views

What are the right categories of finite-dimensional Banach spaces?

This is inspired partly by this question, especially Tom Leinster's answer. Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who ...
Mark Meckes's user avatar
  • 11.4k
18 votes
3 answers
2k views

Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for $C^k$-...
Jochen Wengenroth's user avatar
18 votes
3 answers
1k views

In which sense the GNS-construction is a functor?

I asked this at mathstackexchange a week ago, without success. I think the Gelfand–Naimark–Segal construction must be a functor in some sense, but I can't find an explicit statement anywhere. Can ...
Sergei Akbarov's user avatar
18 votes
3 answers
4k views

Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric. It is essentially ...
Tobias Diez's user avatar
  • 5,824
18 votes
1 answer
564 views

Is the space of Hankel operators complemented in B(H)?

Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$. Let ...
Yemon Choi's user avatar
  • 25.8k
18 votes
1 answer
3k views

Let a function f have all moments zero. What conditions force f to be identically zero?

Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
Zen Harper's user avatar
  • 1,990
18 votes
3 answers
1k views

Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.

This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes. Let $ \...
Leonard's user avatar
  • 816
18 votes
1 answer
5k views

Unbounded linear operator defined on $l^2$

Let $l^2$ be a Hilbert space of infinite sequences $(z_0, z_1, \cdots)$ with finite $\sum_{i=0}^{\infty} |z_i|^2$. Are there any simple example of unbounded linear opearator $T: l^2 \to l^2$ with $D(...
falagar's user avatar
  • 2,821
18 votes
4 answers
1k views

Who first used the multiplication operator version of spectral theory

This is another history question. Hilbert phrased the spectral theorem in terms of resolutions of the identity. While this remained the form of Stone and von Neumann, they did also have the ...
Barry Simon's user avatar
18 votes
1 answer
1k views

Who introduced the notion of "stability" in numerical analysis?

I am preparing a lecture course on the applications of operator theory where I intended to make some numerical analysis application. I was wondering about this question while browsing the literature I ...
András Bátkai's user avatar
18 votes
1 answer
3k views

How bad can the second derivative of a convex function be?

One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property: $$\label{p}\tag{P} f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
Tomás's user avatar
  • 409
18 votes
1 answer
748 views

Banach-Mazur distance between the cube and the octahedron

The Banach-Mazur distance $d(X, Y)$ between two normed spaces $X, Y$ of the same dimension is defined as $d(X, Y) = \log\inf \|T\| \cdot \|T^{-1}\|$, where the $T:X \to Y$ is a linear and invertible ...
tkobos's user avatar
  • 243
18 votes
4 answers
1k views

Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
Manny Reyes's user avatar
  • 5,407
18 votes
2 answers
1k views

compact-open topology on $B(H)$

In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces. Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear ...
André Henriques's user avatar
18 votes
2 answers
776 views

What is known about the "unitary group" of a rigged Hilbert space?

Suppose that $(E,H)$ is a rigged (infinite dimensional, separable) Hilbert space, i.e. $H$ is a Hilbert space, and $E$ is a Fréchet space, equipped with a continuous linear injection $E \rightarrow H$ ...
Peter's user avatar
  • 556
18 votes
3 answers
1k views

Example of a space for which $V \cong Hom(V,V)$

Let $V$ be a topological linear space, and let $\operatorname{Hom}(V,V)$ be the space of continuous linear maps from $V$ back to $V$, equipped with a suitable topology. Is there a non-trivial ...
Tom LaGatta's user avatar
  • 8,512
18 votes
1 answer
2k views

Borel Lemma for vector-valued functions

The classical Borel Lemma states that for an arbitrary sequence $(v_n)_{n \in \mathbb{N}_0}$ of complex numbers there is a smooth function $f\colon \mathbb{R} \longrightarrow \mathbb{C}$ with Taylor ...
Stefan Waldmann's user avatar
18 votes
1 answer
1k views

Commuting unitaries

Is the following true: For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$ there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...
Kate Juschenko's user avatar
18 votes
1 answer
11k views

Is every continuous function measurable?

This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow. In non-Hausdorff topology it is standard to ...
polmath's user avatar
  • 321
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
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

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