All Questions
13,926 questions
19
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2
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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 ...
- 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 $...
- 965
19
votes
1
answer
772
views
convexity of images of space-filling curves
Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t \...
19
votes
2
answers
806
views
Simple connectedness under a metric undistortion condition: on a tricky point in an argument of Gromov
The context
I have been reading Gromov's Metric Structures..., and came upon result 1.14.(a), page 11, which states the following.
Let $K\subset\mathbb R^d$ be a compact subset, and $d_\ell$ its ...
- 3,669
19
votes
1
answer
977
views
Topological universal algebra: what is a variety?
Very roughly, universal algebra is the study of those classes of algebraic structures which can be defined via a set of equations; such a class is called a variety. Of course there is far more to the ...
- 20.5k
19
votes
2
answers
1k
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Existence of continuous map on real numbers with dense orbit?
Does there exist a continuous map $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the forward orbit of 0 is dense in $\mathbb{R}$?
- 189
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)...
- 3,984
19
votes
2
answers
804
views
Existence of a *really* nice topology on the powerset of a topological space
TL;DR. Given a topological space $X$, is there a natural way to "induce" a topology on $\mathcal{P}(X)$ from the topology of $X$ in such a way that 1) all the basic operations of set theory (...
- 11.8k
19
votes
1
answer
1k
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What if homotopy were expanded to allow any connected space instead of $[0,1]$?
What would happen to homotopy theory if we used a more general definition of homotopy, based on general connected spaces rather than $[0,1]$?
Given continuous $f,g:X\to Y$, define $f$ and $g$ to be C-...
- 2,585
19
votes
1
answer
657
views
A large separable space of countable tightness
Is there a ZFC example of a Tychonoff space $X$ such that:
$X$ is separable.
$X$ has countable tightness (that is, a subset of $X$ is closed if and only if it contains the closure of each one of its ...
- 4,413
19
votes
1
answer
465
views
Large Borel antichains in the Cantor cube?
Let $2^\omega$ be the Cantor cube $\{0,1\}^\omega$, endowed with the standard compact metrizable topology and the standard product measure, called the Haar measure. The Cantor cube is considered as a ...
- 41.9k
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 $(\...
- 60.5k
19
votes
0
answers
553
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
563
views
What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$?
Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as ...
- 20.5k
19
votes
0
answers
937
views
What is the Cantor-Bendixson rank of the space of first order theories?
Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...
- 2,146
19
votes
0
answers
703
views
The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?
For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
- 3,104
18
votes
5
answers
2k
views
Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?
I'm curious about the following:
Is every real $n$-manifold isomorphic to a quotient of $\mathbb{R}^n$?
Thanks.
EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking ...
- 761
18
votes
8
answers
2k
views
Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?
What are some of the difficult concepts in topology that have been transferred to graph theory and combinatorics where a certain new application has been found.
A good example is Lovász's proof of ...
18
votes
7
answers
2k
views
Superfluous definitions
It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative.
For if a and b are elements of R, and writing + for the group operation then applying ...
18
votes
4
answers
4k
views
Why are topological ideas so important in arithmetic?
For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in ...
- 4,351
18
votes
3
answers
2k
views
Does Riemann map depend continuously on the domain?
I've always taken this for granted until recently:
In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations $\...
- 375
18
votes
4
answers
3k
views
Generalized Stokes' theorem
In the Wikipedia article on Stokes' theorem the following claim is advanced without any references given:
The main challenge in a precise statement of Stokes' theorem is in defining the notion of a ...
- 181
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 ...
- 1,305
18
votes
3
answers
2k
views
Are finite spaces a model for finite CW-complexes?
Are finite topological spaces (i.e. topological spaces whose underlying set is finite) a model for the homotopy theory of finite simplicial sets (= homotopy theory of finite CW-complexes) ?
Namely, ...
- 43.2k
18
votes
2
answers
1k
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Example of a compact homogeneous metric space which is not a manifold
A metric space $(X,d)$ is isometrically homogeneous if its isometry group acts transitively on points, i.e., for every $x,y \in X$ there is an isometry $\varphi:X\to X$ with $\varphi(x) = y$. I'd ...
- 11.4k
18
votes
3
answers
2k
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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 ...
- 589
18
votes
3
answers
3k
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What are parabolic bundles good for?
The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...
- 2,751
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 ...
- 12.7k
18
votes
2
answers
2k
views
Generalization of Darboux's Theorem
Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$.
Does any of the following generalizations
Let $U\...
- 2,933
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 ...
- 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$-...
- 16.4k
18
votes
2
answers
3k
views
Example of a weak Hausdorff space that is not Hausdorff?
I've looked on the web and haven't found a simple example.
- 728
18
votes
2
answers
2k
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Which platonic solids can form a topological torus?
8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons.
Is the same possible with the ...
- 453
18
votes
2
answers
2k
views
Two definitions of Lebesgue covering dimension
Maybe this question has already been considered here, but after a quick search I didn't find what I was looking for.
As I see, in the literature there are two different definitions of the ...
- 423
18
votes
1
answer
991
views
Is the Robertson–Seymour theorem equivalent to the compactness of some topological space?
The Robertson–Seymour theorem concerns downwardly closed classes of isomorphism classes of finite undirected graphs. (Am I committing some sin by referring to a class of classes? An isomorphism class ...
18
votes
3
answers
5k
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when is a locally homeo a covering map?
Let $X$ and $Y$ be locally comapct Hausdorff spaces, and $f:X\to Y$ be a surjective local homeomorphism.
When is $f$ a covering map?
It is well-known that when $f$ is proper, $f$ is a covering map.
...
- 393
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 ...
- 7,426
18
votes
2
answers
630
views
Is the notion of fixed point property for topological spaces an absolute notion?
Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More ...
- 32.2k
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 ...
- 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 ...
- 25.8k
18
votes
1
answer
3k
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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 ...
- 1,990
18
votes
1
answer
3k
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Proper discontinuity and existence of a fundamental domain
I am currently teaching a topics course where I talk about some discrete groups acting properly. A student asked a very basic question that stumped me: what is the precise relationship between proper ...
- 1,574
18
votes
3
answers
1k
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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 $ \...
- 816
18
votes
1
answer
5k
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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(...
- 2,821
18
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4
answers
1k
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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 ...
- 827
18
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3
answers
1k
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Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?
Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space?
Here is one natural ...
- 8,512
18
votes
1
answer
1k
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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 ...
- 4,729
18
votes
2
answers
1k
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The Klein bottle and the Heawood Conjecture
Let $\Sigma_g$ be a surface of genus $g$. The Heawood Conjecture gives a closed formula in one variable, $\chi$ (the Euler characterstic of $\Sigma_g$), for the minimal number of of colors needed to ...
- 1,180
18
votes
1
answer
3k
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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,...
- 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 ...
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