All Questions
13,927 questions
13
votes
3
answers
1k
views
Map from simplex to itself that preserves sub-simplices
I believe this may be a standard algebraic topology problem, so I apologize in advance if this belongs in stackexchange (it's not a homework problem, however, and came about in a research context). I'...
- 645
13
votes
3
answers
2k
views
A quotient space of complex projective space
Let $\mathbb{C}P^n$ be the $n$-dimensional complex projective space and denote $[z_0:\dots:z_n]$ its points. If we glue $[z_0:\dots:z_n]$ and $[\overline{z_0}:\dots:\overline{z_n}]$ for any $[z_0:\...
- 331
13
votes
3
answers
820
views
Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?
This question is related to another one that I asked two days ago.
Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with
the following two properties?
The ...
- 1,396
13
votes
4
answers
5k
views
What is known about the Gaussian measure of the unit ball in a Hilbert Space?
Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ ...
- 617
13
votes
3
answers
2k
views
Sobolev spaces and geometry
This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study PDE'...
- 947
13
votes
2
answers
1k
views
Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
13
votes
2
answers
915
views
Topological vector spaces (reference request)
In his book Topological Function Spaces Arhangel'skii says that "it is well known that every nontrivial locally convex linear topological space $X$ is homeomorphic to a space of the form $Y \...
- 674
13
votes
6
answers
3k
views
When does local invertibility imply invertibility?
Generally, local invertibility does not imply invertibility. However, for differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ then surjectivity and local invertibility do imply invertibility.
...
- 26.8k
13
votes
4
answers
2k
views
Is the category of Banach spaces with contractions an algebraic theory?
Consider the category of Banach spaces with contractions as morphisms (weak, so $\|T\| \le 1$). Is this an algebraic theory?
I suspect that this is true. The "operations" will be weighted sums, ...
- 26.8k
13
votes
2
answers
2k
views
When can we divide continuous functions?
Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$.
What can be said ...
- 5,529
13
votes
3
answers
2k
views
Space of sections of a fibre bundle with non-compact base space
Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$.
For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9)...
- 5,824
13
votes
1
answer
545
views
Square of a continuous map
Recently a student asked me the following (elementary looking) question :
If $T$ is an invertible linear transformation of some finite-dimensional space $E$ into itself which factorizes as $T = f \...
- 7,249
13
votes
3
answers
2k
views
A characterization of $L_1(\mu)$ in $L_\infty(\mu)^*$
Let $\mu$ be a finite positive measure on a set $M$:
$$
\mu(M)<\infty.
$$
As is known, the Banach dual space $L_\infty(\mu)^*$ to the space $L_\infty(\mu)$ contains $L_1(\mu)$, but (excluding some ...
- 7,426
13
votes
3
answers
2k
views
The "miracle" of Heegard Floer.
Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins ...
- 804
13
votes
7
answers
10k
views
What is the best reference for Spectral theory?
I'm studying Bernard Aupetit: A Primer on Spectral Theory
but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things?
Thank you.
- 823
13
votes
1
answer
2k
views
Analysis of the boundary of the Mandelbrot set
Motivation: The Mandlebrot set is a simply connected set with an infinitely complex boundary, but CAN one move from interior to the exterior of this topological space by just crossing over a finite ...
- 851
13
votes
1
answer
662
views
Frankl's conjecture restricted to finite topological spaces
A finite topological space is a finite family of finite sets that is closed under both union and intersection.
Frankl's conjecture states that for any finite union-closed family of finite sets, ...
- 2,649
13
votes
2
answers
646
views
Functions separting points in Hausdorff spaces
A colleague in algebra asked me this, and I couldn't answer it. On the Wikipedia page for "epimorphism" it is claimed that in the category of Hausdorff spaces and continuous maps, a function is epi ...
- 18.7k
13
votes
1
answer
592
views
Topological semi-direct products of groups
In Kaniuth, Taylor, Induced representations of locally compact groups on pages 9-10 it's claimed that if $G$ is a locally compact group with closed subgroups $N,H$, with $N$ normal in $G$, with $N\cap ...
- 18.7k
13
votes
1
answer
728
views
Explicit isomorphism $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_1(\mathbb{RP}^{n-1})$
From covering space theory we know that $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_{n+1}(\mathbb{S}^n)$.
From wikipedia I can notice that $\pi_{n+1}(\mathbb{S}^n) \cong \pi_1(\mathbb{RP}^{n-1})$.*
My ...
- 1,263
13
votes
4
answers
1k
views
nonhausdorff dimension
if $X$ is a topological space, a first step in making $X$ hausdorff is taking the quotient $H(X)=X/\sim$, where $\sim$ is the equivalence relation generated by: if $x,y$ cannot be seperated by ...
- 63.1k
13
votes
2
answers
767
views
Smooth Urysohn's lemma on Fréchet spaces
Let $V$ be a Fréchet topological vector space.
Let $K_0$ and $K_1$ be two closed subsets which are disjoint.
I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$
whose restriction ...
- 43.2k
13
votes
1
answer
639
views
$T_2$-spaces where all non-empty open sets are homeomorphic
We say that a $T_2$-space $(X,\tau)$ has homeomorphic open sets if every non-empty open set $U\subseteq X$ endowed with the subspace topology is homeomorphic to $(X,\tau)$.
The rationals with the ...
- 48.9k
13
votes
2
answers
897
views
Can non-central projections still commute with all other projections?
Let $A$ be a C*-algebra and let $\mathcal{P}(A)$ denote the set of projections in $A$. If $p\in\mathcal{P}(A)$ commutes with everything in $\mathcal{P}(A)$ does it necessarily commute with everything ...
- 1,307
13
votes
1
answer
442
views
Is the identity function a unique multiplicative homeomorphism of $\mathbb N$?
Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N_0$ where $\mathbb N_0=\{0\}\cup\mathbb N$, where $a,b\...
- 42k
13
votes
1
answer
797
views
Is there a compact, connected, totally path-disconnected topological group?
There exist homogeneous spaces such as the pseudo-arc, which are compact, connected, and totally path-disconnected. Is there a nontrivial, Hausdorff topological group with the same properties, i.e. ...
- 7,193
13
votes
3
answers
1k
views
Is the set of separable quantum states closed?
Let $\mathcal H,\mathcal H'$ be Hilbert spaces (not necessarily separable).
A "separable state" is a trace-class operator of the form $\sum_i \rho_i\otimes\rho_i'$ where $\rho_i,\rho_i'$ are positive ...
- 896
13
votes
5
answers
1k
views
Connectedness in the plane
There are several open problems in topology which concern connectedness and subsets of the plane. The biggest of these is undoubtedly:
Question. Does every non-separating plane continuum have the ...
- 955
13
votes
1
answer
911
views
Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic?
It's easy to see that, for $1\le p,q< \infty$ the spaces $L^p(\Bbb R)$ and $L^q(\Bbb R)$ of $p$-th and $q$-th power integrable functions on the real line are homeomorphic as topological spaces. In ...
- 3,017
13
votes
2
answers
1k
views
Applications of non-separable Hilbert spaces
In applications, Hilbert spaces of interest are often assumed to be separable. In addition to being extremely convenient mathematically, this assumption can often be justified on computational or ...
13
votes
2
answers
4k
views
Structure theorem for finite dimensional $C^*$-algebras and their representations
I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere.
Let $\mathcal{A} \subset M_n(\mathbb{C})$ be an ...
- 1,429
13
votes
3
answers
650
views
General principles which lead to good questions in many concrete situations [closed]
I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such ...
13
votes
1
answer
675
views
Strongly rigid Hausdorff spaces
A space $(X,\tau)$ is called rigid if $\textrm{Aut}(X)=\{\textrm{id}_X\}$. We say $(X,\tau)$ is strongly rigid if for every continuous map $f:X\to X$ we have that $f = \textrm{id}_X$ or $f$ is ...
- 48.9k
13
votes
2
answers
769
views
How hard (P, NP, NP-hard) is it to compute Schur norms of matrices (as multipliers)?
Given a matrix $A\in M_n(\mathbb{C})$, I will denote by $||A||_\infty$ the operator norm of $A$, as seen acting on the Hilbert space $\mathbb{C}^n$. This makes $M_n(\mathbb{C})$ into a Banach space (...
- 647
13
votes
1
answer
3k
views
metric on the space of real analytic functions
Hello,
this question may be simple but I couldn't find a reference.
Let $E$,$F$ be real Banach spaces and $\Omega\subset E$ be a bounded domain and let $C_b^{\omega}(\Omega,F)$ be the vector space of ...
- 223
13
votes
1
answer
576
views
Are Hausdorff measures on the real line Haar measures for some locally compact topology?
For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\...
- 32.5k
13
votes
1
answer
947
views
When is $X \rightarrow \text{Spec}(C(X))$ a homeomorphism?
Let $X$ be compact Hausdorff topological space. Consider the ring $C(X)$ of continuous functions $X \rightarrow \mathbb C$ (we do not consider the C* algebra structure, just consider $C(X)$ as a ring) ...
- 11.9k
13
votes
3
answers
555
views
If G is a sequential topological group, must G x G be sequential?
Using standard definitions, the topological space $Y$ is sequential if for each nonclosed $A \subset Y$, there exists a convergent sequence $a_{1}$ , $a_{2}$,...$\rightarrow b$
so that $a_{n} \in A$ ...
- 1,968
13
votes
5
answers
1k
views
Does this sequence span $L^2$?
Consider the following sequence of functions in $L^2[0,\infty)$:
$$f_n(x)=e^{-x/n}x^n,\;\;n\geq 1$$
Does this sequence span $L^2[0,\infty)$ (that is, is the set of finite linear combinations
of these ...
- 520
13
votes
1
answer
766
views
Is Top_4 (normal spaces) a reflective subcategory of Top_3 (regular spaces)?
I’m studying some category theory by reading Mac Lane linearly and solving exercises.
In question 5.9.4 of the second edition, the reader is asked to construct left adjoints for each of the inclusion ...
- 1,411
13
votes
3
answers
671
views
How algebraic can the dual of a topological category be?
(I'm going to try to use definitions from Abstract and Concrete Categories: The Joy of Cats by Adámek, Herrlich, and Strecker, since both of the adjectives in the title of my question seem to have at ...
- 13k
13
votes
2
answers
653
views
The geometry of $\mathbb{R}^n$
Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces.
Then we define the set of equivalence classes
$$G(X,Y):=\left\{[T]; T,S \in ...
- 536
13
votes
1
answer
1k
views
An inequality for the spectral radius of matrices used by J. Bochi
I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
- 6,206
13
votes
1
answer
1k
views
Between compact and locally uniform: What is the name of this convergence?
Let $X$ be a topological space, $(Y,d)$ a metric space, $f\in Y^X$, and $(f_n)$ a sequence in $Y^X$ with the following property:
For every $x_0\in X$ and every $\varepsilon>0$, there exist a ...
- 802
13
votes
1
answer
863
views
Mistake on article about Bohr compactification?
$\DeclareMathOperator\b{b}\newcommand\B{{\operatorname B}}$I wish to get help understanding the content of two theorems of [Iva] that seem mutually contradictory. First some context. Let $\b(\mathbb{R}...
- 193
13
votes
1
answer
980
views
Any "natural" topology on fractional field of a topological ring?
Let $R$ be a topological integral domain. Let $K=\mathrm{Frac} R$. Is there any "natural" topology on $K$? Actually, since $K$ can be regarded as a quotient of $R\times R$ quotient some equivalence ...
- 153
13
votes
2
answers
2k
views
What is the relationship amongst all the different kinds of spectra?
The word "spectrum" gets tossed around a lot in mathematics, and there seem to be a number of different concepts to which it applies. There is of course a physical connotation to the word which is ...
13
votes
2
answers
570
views
A conjecture of De Giorgi on weighted Sobolev spaces
Let $\mu$ be a probability measure on $\mathbb{R}^d$ which is absolutely continuous with respect to the Lebesgue measure with density $\rho$. Assume that, for all $t>0$,
\begin{align*}
\exp \left(...
- 778
13
votes
2
answers
552
views
Existence of closed operators with arbitrary dense domain of a given Banach space
Consider any Banach space $X$, and let $Y$ be any dense subspace, then does it necessarily exist a closed linear operator $T$ defined on $X$, such that the domain of $T$ is exactly $Y$, i.e., $D(T)=Y$?...
- 879
13
votes
1
answer
558
views
Idempotent ultrafilters and the Rudin-Keisler ordering
Short version: what can we say about the place of idempotent ultrafilters in the Rudin-Keisler ordering?
Longer version:
If $U$, $V$ are (nonprincipal) ultrafilters on $\omega$, then we write $U\ge_{...
- 20.5k