All Questions
4,469 questions with no upvoted or accepted answers
77
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\...
- 4,060
46
votes
0
answers
2k
views
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$...
- 42.8k
43
votes
0
answers
820
views
A kaleidoscopic coloring of the plane
Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
- 41.8k
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-...
- 25.5k
31
votes
0
answers
2k
views
Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?
Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $\|x\| = 1$ and $\|Ax\| = \|A\|$. The ...
- 4,878
31
votes
0
answers
1k
views
When are two C*-algebras isomorphic as Banach spaces?
We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its ...
- 3,497
30
votes
0
answers
899
views
Three real polynomials
Theorem. Let $f,g$ be two real polynomials, and suppose that their Wronskian $W(f,g)=f'g-fg'$ has only real roots. Then on any interval $I\subset\mathbf{R}$ containing no roots of $W$ every non-...
- 91.8k
28
votes
0
answers
1k
views
Number of real roots of a polynomial
Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$. Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b)...
- 6,476
27
votes
1
answer
1k
views
The dual of $\mathrm{BV}$
$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
- 683
27
votes
0
answers
1k
views
Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 356
23
votes
0
answers
939
views
A question about small sets of reals
In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, we have that $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is ...
- 9,631
23
votes
0
answers
1k
views
Laplace Transform in the context of Gelfand/Pontryagin
Questions:
Is there a class of objects (presumably related to locally compact abelian groups) for which the quasi-characters canonically generalize the Laplace transform?
If not, is there a ...
- 1,104
21
votes
0
answers
658
views
A multiple integral
Let us consider the multiple integral
$$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots
\int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots
\cos {(s_{2n-1}...
- 16.5k
21
votes
0
answers
869
views
Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
- 28.6k
21
votes
0
answers
1k
views
Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$
A while ago, I came across the following problem, which I was not able to resolve one way or the other.
Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and $g(t,...
- 17.1k
21
votes
0
answers
732
views
Closed connected additive subgroups of the Hilbert space
It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...
- 60.5k
21
votes
0
answers
876
views
Are the eigenvalues of the Laplacian of a generic Kähler metric simple?
It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
- 6,247
20
votes
0
answers
634
views
Is $\sum_{n=1}^\infty \frac{n!}{n^n}$ rational?
Is $\displaystyle \sum_{n=1}^\infty \frac{n!}{n^n}$ rational?
This question has been posted in MSE for two years without an answer. A094082 seems to suggest that it is not rational. Is it still an ...
user14319
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{...
- 484
19
votes
0
answers
775
views
A Linear Order from AP Calculus
In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that $...
- 433
18
votes
0
answers
1k
views
Does there exist a continuous open map from the closed annulus to the closed disk?
(Originally from MSE, but crossposted here upon suggestion from the comments)
In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
- 831
18
votes
1
answer
2k
views
Function of two sets intersection
Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
- 1,209
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 ...
- 37.8k
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\...
- 4,878
16
votes
0
answers
542
views
$C^*$-algebra generated by those operators that are bounded on every $\ell_p$
Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...
- 25.8k
16
votes
0
answers
1k
views
Finite Rank Commutators
My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...
- 31.5k
15
votes
0
answers
244
views
Natural examples of Borel surjections without right inverse
As discussed in this question, in general a Borel surjection $f:\mathbb{R}\rightarrow\mathbb{R}$ may not have a Borel right inverse, namely a $g$ such that $f\circ g=id$, although there is always a ...
- 623
15
votes
1
answer
601
views
Topological spaces in which countable intersections of dense open sets have dense interior
In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense.
Now consider the following strengthening of the Baire ...
- 32.5k
15
votes
0
answers
477
views
Quantitative Skorokhod embedding
The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
- 723
15
votes
0
answers
749
views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
- 178
15
votes
0
answers
409
views
Is there a continuous map $f:\mathbb R^\omega\to\mathbb R^\omega$ with dense countable preimage $f^{-1}(\mathbb Q^\omega)$?
Let $\mathbb Q^\omega_0:=\{(x_i)_{i\in\omega}\in\mathbb Q^\omega:\exists n\in\omega\;\forall m\ge n\;\;x_m=0\}$ and observe that $\mathbb Q^\omega_0$ is a countable dense set in $\mathbb R^\omega$ (...
- 41.8k
15
votes
0
answers
510
views
Lebesgue density 1/2 (or bounded away from 0 and 1)
From the work of Preiss, we know that in infinite-dimensional spaces, one has violations of the Lebesgue density theorem. In particular, he has constructed examples of probability spaces where a set ...
- 6,541
15
votes
0
answers
365
views
Admissible relations in a Banach algebra
Suppose that $\mathbb{C}\left\langle x, y \right\rangle = R$ is a free (associative and unital) algebra and $f \in R$. I wonder whether there exists a (unital) Banach algebra $A$ and a non-zero pair $...
- 193
15
votes
0
answers
349
views
Is there support for the term "Gelfand algebra"?
In this question Yemon Choi asked whether there is a standard term for Banach algebras for which the submultiplicative law
($\|ab\| \leq \|a\| \|b\|$) is weakened to merely requiring the product to be ...
- 42.8k
15
votes
0
answers
1k
views
Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle
My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...
- 648
14
votes
0
answers
860
views
strong topologies on $C_c^\infty$
UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
- 486
14
votes
0
answers
718
views
Lower bounds on analytic functions connected to Fox H
The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
- 178
14
votes
0
answers
205
views
Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?
A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article:
W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$.
In Convex ...
- 25.8k
14
votes
0
answers
633
views
Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
- 24.7k
14
votes
0
answers
3k
views
Tanh version of a Fourier Transform?
I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...
- 3,979
14
votes
0
answers
2k
views
Schwartz kernel theorem for A-linear operators
Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^\infty(X), ...
- 7,637
13
votes
0
answers
710
views
Minimizing total variation under constraint
For $p\in[0,1]$, we write $\mathrm{Ber}(p)$
to denote the Bernoulli measure on $\{0,1\}$;
that is, $\mathrm{Ber}(p)(0)=1-p$,
$\mathrm{Ber}(p)(1)=p$.
For $n\in\mathbb{N}$ and $p=(p_1,\ldots,p_n)\in[0,1]...
- 6,541
13
votes
0
answers
573
views
Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry
In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...
- 2,339
13
votes
0
answers
174
views
Existence of more than two C*-norms on algebraic tensor product of C*-algebras
Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$.
If $A$ or $B$ is nuclear, then all pairs $(A,B)$...
- 536
13
votes
0
answers
545
views
Is there a logical relationship between constructions of the reals and proof methods in real analysis?
In my elementary real analysis course three years ago, I remember noting that there seemed to be 3 main ways of proving the main theorems about continuity. There was Bolzano-Weierstrass, continuous ...
- 295
13
votes
0
answers
818
views
Covering number estimates for Hölder balls
Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
- 5,405
13
votes
0
answers
491
views
Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?
Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...
- 82.7k
13
votes
0
answers
395
views
Converse to Riesz-Thorin Theorem
Let $T$ be an operator on simple functions on (say) $\mathbb{R}$.
The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$....
- 854
13
votes
0
answers
372
views
Finite dimensional approximation of Donaldson theory
In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...
- 17.5k
13
votes
0
answers
324
views
Banach spaces with $d(X,Y) = 1$
We recall that the Banach-Mazur distance between two isomorphic Banach spaces is given by $d(X,Y) = \inf \{ \|T\| \|T^{-1}\| : T$ is an isomorphism from $X$ to $Y\}$.
It is a classical result that we ...
- 538