All Questions
1,591 questions
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Weak* continuity of positive parts
I'm a little embarrassed to be asking this, but surely there is a simple argument that I didn't see?
Let $(f_\lambda)$ be a net in $l^\infty$ which converges weak* to $f \in l^\infty$. We do not ...
- 42.8k
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1
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Separating vectors for C$^*$-algebras
(I asked this on math.stackexchange, without response).
Let $A$ be a C$^*$-algebra, concretely acting on a Hilbert space $H$. Suppose that $\xi_0\in H$ is cyclic and separating for $A$ (that is, the ...
- 18.7k
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Quantum functional analysis
Can one explain some philosophy behind "quantum functional analysis" (or "quantized functional analysis") which was initiated and developed by such researchers as: Ruan Z.-J., Pisier J., Effros E.G., ...
- 537
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0
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657
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“Taylor series” is to “Volterra series” as “Laurent series” is to _________?
Preamble
My question is similar to an earlier MathOverflow question:
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________? which I just answered (hopefully my first ever ...
- 446
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2
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1k
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On equibounded sequences in $L^\infty$
Let $f_n: [0, 1] \to \mathbb R$ be a sequence of positive functions in $L^\infty$ (hence a fortiori in $L^1$) that are equibounded in $L^\infty$ norm - that is $\sup_{n \in \mathbb N} \|f_n\|_{L_\...
- 6,223
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422
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Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
- 825
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417
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A variant of the Moore-Mrowka problem
A space $X$ is said to be sequential if whenever $A \subset X$ is not closed then $A$ contains a sequence converging to a point outside of $A$.
A space $X$ is said to have countable tightness if for ...
- 4,413
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2
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750
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Is there a compact space with no countably generated dense subspace?
This is a reformulation of this MO question which recieved little or no attention due to the fact that the OP gave no motivation whatsoever. I found the question quite interesting and decided to give ...
- 11.5k
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477
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Is $\mathbb{R}^\omega \cong (\mathbb{R}^\omega \setminus \{x\})$?
Is $\mathbb{R}^\omega \cong (\mathbb{R}^\omega \setminus \{x\})$, where $\mathbb{R}^\omega$ is given the product topology, and $x\in\mathbb{R}^\omega $?
- 48.9k
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Counting norms on an infinite dimensional vector space
It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...
- 101
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2
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606
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A characterization of metric spaces admitting a bi-Lipschitz embedding into a Hilbert space?
Theorem (??) derived in this MO-post from Schoenberg's theorem yeilds a "bipartite" characterization of metric spaces that admit an isometric embedding into a Hilbert space. This Theorem (??)...
- 41.9k
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901
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Approximation of a compactly supported function by Gaussians
Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
- 710
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What is known about topological groups of countable spread in ZFC?
A topological space has countable spread if every discrete subspace is at most countable.
By Theorem 8.10 in Todorcevic's book "Partition Problems in Topology", PFA implies that each regular space $X$...
- 41.9k
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4
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References: Infinite dimensional Lie algebras
What I really want are properties (if it is abelian, nilpotent, solvable, simple, or semisimple; Cartan subalgebras...) of the Lie algebra of smooth functions on a symplectic manifold $(M,\omega)$; ...
- 641
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595
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Density of smooth function in Hilbert spaces
I am looking for a simple reference to the following fact:
If $f:\Omega\to\mathbb{R}$ is continuous, where $\Omega\subset H$ is an open subset of a separable Hilbert space $H$, then for any $\...
- 28k
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2
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666
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Reference request: Extensions of Wiener's Tauberian Theorem
Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
- 710
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3
answers
2k
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Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
- 360
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381
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Why is this space contractible?
Is the following space, obtained by glueing a Cantor set worth of "hairs" to a closed disk in $\Bbb R^2$ contractible?
The obvious attempt of contracting the hairs to the root and then ...
10
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1
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594
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Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?
Added. My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails ...
- 13.8k
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3
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2k
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Pathological product space norm
Let $X$ and $Y$ be two normed vector spaces and $n(\cdot, \cdot)$ be any norm on $\mathbb{R}^2$. Is it always possible to define a norm on the product vector space $X \times Y$ as $||(x, y)||_{X \...
- 202
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423
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a question about connected open sets in $R^2$
Let $U,V$ be two nonempty connected open sets in $\mathbb{R}^2$ and $U\subsetneqq V$.I want to ask if there must exist an open ball $B\subset V$ such that $B\not\subset U$ and $B\cap U$ is a nonempty ...
- 1,997
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598
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What happens if we rotate the kernel of an integral operator?
Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...
- 452
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452
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Quotient of $S^3$ by Montgomery and Zippin's "wild involution"
In 1952, Bing showed the existence of a topological involution of $S^3$ with fixed point set the Alexander horned sphere, demonstrating that $S^3$ has finite-order homeomorphisms not conjugate to ...
- 2,851
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270
views
Which compact metrizable spaces have continuous choice functions for non-empty closed sets?
Let $X$ be a compact metrizable space and let $\mathcal{K}_{ne}(X)$ be the collection of non-empty closed subsets of $X$ with the Vietoris topology (i.e. the topology induced by the Hausdorff metric ...
- 12.5k
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695
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Topology from the viewpoint of the filter endofunctor
Question. Are there any references that develop general topology from the viewpoint of a functor $$\Phi : \mathbf{Rel} \rightarrow \mathbf{Rel}$$ that assigns to every set $X$ the set $\Phi(X)$ of ...
- 3,793
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314
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How much do idempotent ultrafilters generate in terms of semigroups?
It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
- 1,914
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2
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711
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Mapping space from a quotient space
For $X/{\sim}$ a quotient space,
$$
Map(X/{\sim},Y)\subset Map(X,Y).
$$
But is this inclusion always a homeomorphism on its image? (Assuming compact-open topology on the mapping spaces.) If not what ...
- 1,875
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207
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Projective tensor squares of uniform algebras
In discussion with a colleague recently (Jan 2017),
$\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$
I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
- 25.8k
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226
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Extremal bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ for a Banach space $X$ is called extremal if there exists a point $s$ in the unit sphere $S_X=\{x\in X:\|x\|=1\}$ such that for every $i\in\{1,\dots,n\}$ the ...
- 4,535
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350
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What is the smallest density of a metrizable space without countable separation?
A Tychonoff space $X$ is defined to have countable separation if some (equivalently, any) compactification $bX$ of $X$ contains a countable family $\mathcal U$ of open sets such that for any points $x\...
- 41.9k
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2
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739
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Unconditionally convergent series in some functional spaces
Linked with this question and discussion
(Bilinear product of two summable families), I am very
interested in counterexamples/results about the following questions (cf the end).
First, I recall that a
...
- 3,738
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1
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658
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Are functions of moderate growth a bornological space?
I was thinking a bit about distribution theory the last weeks and stumbled across the following question:
There are two natural locally convex topologies on the space of smooth functions of moderate ...
- 9,650
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Is a space with no covering spaces simply connected?
Suppose $X$ is a path connected space such that every connected covering space of $X$ is trivial (1-fold.) Must $X$ be simply connected?
Intuitively, the answer seems to be no (imagine taking a disk,...
- 713
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242
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Arhangel'skii's problem revisited
One of the most well-known problems in set-theoretic topology is Arhangel'skii's question of whether there exists a Lindelöf Hausdorff space with "points $G_\delta$" (meaning, every point is ...
- 4,413
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441
views
A new $\ell_p$-metric on the hyperspace of finite sets?
Let $(X,d)$ be a metric space and $Fin(X)$ be the family of all non-empty finite subsets of $X$. For every $n\in\mathbb N$ the elements of the power $X^n$ are thought as functions $f:n\to X$ where $n:=...
- 41.9k
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2
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244
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Minimal refinements of open covers of $T_2$-spaces
Let $(X,\tau)$ be a topological space. We say ${\cal U}\subseteq \tau$ is an open cover if
$\bigcup {\cal U} = X$, and
$X\notin {\cal U}$.
${\cal U}$ is minimal if for all $U_0\in {\cal U}$ we have ...
- 48.9k
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Why does Riesz's Representation Theorem apply in quantum mechanics?
$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$.
It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
- 2,071
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514
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Cofinal monotone maps from $\omega^\omega$ to $\kappa^\kappa$
Given a cardinal $\kappa$ consider the set $\kappa^\kappa$ of all functions from $\kappa$ to $\kappa$, endowed with the partial order $f\le g$ iff $f(\alpha)\le g(\alpha)$ for all $\alpha\in\kappa$.
...
- 41.9k
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531
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Existence of infinite groups that are too reluctant to be topological
With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...
- 1,145
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726
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Uniform Embedding into Euclidean Space
Given a locally compact, separable, metric space $X$.
When does $X$ uniformly embed into some Euclidean space?
This means, when does there exist some integer $n$ and a closed subset $Y\subset\...
- 3,497
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211
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Is the category of all topological spaces, including the bad ones, simplicially tensored and cotensored?
Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones.
We can make $\textbf{Top}$ into a simplicially enriched category as follows:
Given topological spaces $X$ and $Y$,...
- 15.9k
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797
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Lorenz attractor path-connected?
Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure.
EDIT: The answer below is unsatisfactory, and possibly ...
- 331
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390
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Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?
According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point.
However, these fixed points cannot be chosen ...
- 2,130
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226
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Is $\beta\mathbb N$ a unique compactification with the smallest possible permutation group?
For a compactification $c\mathbb N$ of $\mathbb N$ let $\mathcal H(c\mathbb N,\mathbb N)$ be the group of homeomorphisms $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\...
- 41.9k
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596
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Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?
Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum
$$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$
Now ...
- 335
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338
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Commuting nets for commuting projections
I think this should not be too difficult, but I am not an expert. I did not get an answer on stackexchange.
Let $A$ be a $C$*-algebra and let $p,q\in A^{**}$ be two commuting projections. Then there ...
- 633
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4
answers
1k
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Boundedness of nonlinear continuous functionals
Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$.
Is it true that there exists an infinite dimensional reflexive subspace
$E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ?
...
- 4,060
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1
answer
336
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How much can complexities of bases of a "simple" space vary?
Given a countable subbase of a topology, we can consider its complexity in terms of the difficulty of determining whether one family of basic open sets covers another basic open set. My question is ...
- 20.5k
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758
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Convergence of Schwartz kernels implies convergence of operators
Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
- 2,998
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4
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2k
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How may I find all continuous and bounded functions g with the following property?
Find all continuous and bounded functions $g$
with :
$$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
I have posted this question here, but received no answer.
- 4,074