All Questions
1,584 questions
8
votes
3
answers
747
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How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]
Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
8
votes
1
answer
485
views
An inequality related to Riesz–Thorin theorem, determinants and $L_p$ norm
Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that
$$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\
a_j & b_j & c_j \\
...
- 563
8
votes
1
answer
716
views
Topological fraction rings and fields
Linked to this question
and as a sequel to my answer of it.
Let $R$ be a topological (commutative, unital) ring and set $S$ be a submonoid of $(R,\times,1_R)$.
Let
$$
s_{frac}\ :\ R\times S\to S^{-...
- 3,738
8
votes
2
answers
488
views
If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable?
Does there exits any non-separable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?
- 4,058
8
votes
1
answer
386
views
Lower bound for $\frac{\sum_{i,j}\min((f_i-f_j)^2,(g_i-g_j)^2)}{\sum_{i,j}\max((f_i-f_j)^2,(g_i-g_j)^2)}$
Let $f\in\mathbb{R}^n$ and $g\in\mathbb{R}^n$ be two orthogonal unit vectors such that $\sum_{i}{f_i}=\sum_{i}{g_i}=0$.
Question. Can we prove this?
$$\frac{\sum_{\{i,j\}}\min((f_i-f_j)^2,(g_i-...
- 519
8
votes
0
answers
182
views
Distribution domination for sums of independent random variables in Banach spaces
Let $X$ be a Banach space and let $(\xi_n)$ and $(\eta_n)$ be independent mean-zero random variables with values in $X$ satisfying
$$
\sum_n \mathbb P(\xi_n \in A) \leq \sum_n \mathbb P(\eta_n \in A),
...
- 131
8
votes
2
answers
1k
views
Trivial fiber bundle
Suppose $(E,p,B;F)$ is a fiber bundle such that $E$ is homeomorphic to $B\times F$, is it true that the fiber bundle is trivial?
A non connected counter example has been provided, so I'll ask for E,B ...
- 2,751
8
votes
1
answer
5k
views
integration by parts for the fractional Laplacian
Is there an integration by parts formula for fractional laplacians in $L^p(\mathbb{R}^N)$, something like
$$
s\in(0,1),\qquad\int\limits_{\mathbb{R}^N}f[(-\Delta)^sg] =\int\limits_{\mathbb{R}^...
- 5,380
8
votes
2
answers
2k
views
End point compactification for metric spaces
Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here).
For ...
- 3,033
8
votes
1
answer
340
views
characterization of commutative Banach algebras
Let $A$ be a Banach algebra with the following property:
For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges.
...
- 356
8
votes
3
answers
1k
views
Fourier dimension of the sum of sets
This question came up when my supervisors, my colleague, and I were considering arithmetic progressions in sets of fractional dimension. In particular, we were interested in "extracting" Salem sets ...
- 505
8
votes
2
answers
785
views
Is taking the product of signed measures weakly continuous?
For a Polish space $X$, let $C_b(X)$ denote the real Banach space of bounded continuous real-valued functions on $X$. Let $M(X)$ denote the space of all finite signed Borel measures on $X$, equipped ...
- 29.8k
8
votes
4
answers
749
views
A question about local connectedness
Let C be a connected and completely metrizable subset of the Euclidean plane. Can C fail
to be locally connected at each of its points?
- 6,215
8
votes
2
answers
415
views
Space $X$ such that $X^\lambda\cong X$ for some $\lambda$
Which cardinals $\lambda > 2$ have the following property?
There is a space $(X,\tau)$ such that
for all cardinals $\kappa$ with $1<\kappa<\lambda$ we have $X\not\cong X^\kappa$, and
$X\...
- 48.9k
8
votes
3
answers
3k
views
The mean of points on a unit n-sphere $S^n$
A unit n-sphere is defined as $$\mathcal{S}^n = \{\mathbf{p} \in \mathbb{R}^{n+1}: \|\mathbf{p}\| = 1\}$$
The distance between two points $\mathbf{p}$, $\mathbf{q}$ on $\mathcal{S}^n$ is the great-...
- 147
8
votes
2
answers
1k
views
What does the unique mean on weakly almost periodic functions look like?
There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...
- 4,432
8
votes
4
answers
812
views
Schwartz space of functions with values in a Frechet space
While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in ...
- 601
8
votes
2
answers
3k
views
$L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$
It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in ...
- 176
8
votes
3
answers
1k
views
Dual Banach space of $B(X,Y)$ when $X$ is finite dimensional
Denote $B(X,Y)$ the Banach space of bounded operators between Banach spaces $X$ and $Y$.
When $X$ and $Y$ are both finite dimensional, it follows from the formula
$$\|u\|_{B(X,Y)} = \sup_{\|x\|_X <...
- 9,486
8
votes
3
answers
485
views
Does the metric space of compact metric spaces satisfy the binary intersection property?
A metric space $Y$ has the binary intersection property provided that whenever a collection of closed balls in $Y$ intersects pairwise, then there is a common intersection point.
Does the metric ...
- 15.5k
8
votes
2
answers
499
views
Refining open covers in locally path connected spaces
Suppose $X$ is a locally path connected topological space and $\mathcal{U}$ is an open cover of $X$ (consisting of path connected sets if we want).
One often wants the intersection $A\cap B$ of ...
- 7,193
8
votes
3
answers
2k
views
Measures on general topological groups
I am interested in the group algebras of non-locally compact groups. What references can you advise?
This is a wide question, so I list more concretely what I would like to see:
Here X can be even ...
- 1,152
8
votes
1
answer
2k
views
Fastest decay of Fourier Transform for Generalized Functions of compact support
What is the fastest decay possible for the Fourier transform of a generalized function with compact support and finite value at the origin? I know that regular functions cannot attain exponential ...
- 83
8
votes
2
answers
3k
views
Bialynicki-Birula decomposition of a non-singular quasi-projective scheme.
Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G_m$ and a non-singular scheme $X$ of finite type over $k.$
Let us define the following:
Condition 1: $X$ can be covered by ...
user17778
8
votes
1
answer
421
views
$C^k$ one-parameter family of metrics
Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
- 1,407
8
votes
4
answers
2k
views
Manifold-Valued Sobolev Spaces
I have the following basic question about Sobolev-spaces which take their values in a
Riemannian manifold $(M,g)$, i.e.
functions $u:\Omega \to M$, $\Omega \subset \mathbb{R}^n$ bounded, such that ...
- 233
8
votes
2
answers
583
views
Does every operator from a Hilbert space to $L^0$ factor through a canonical one?
Let $A:H\to L^0(S, \mu)$ be a continuous operator from a Hilbert space to the space of (equivalence classes of) measurable functions on a probability measure space $S$ with convergence in measure. Let'...
- 4,010
8
votes
2
answers
2k
views
when a pseudo-differential operators to be compact?
In the theory of Pseudo-differential operators,when a symbol $a(x,\xi)\in S^{0}$,then the operator $a(x,D)$ defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$ is $L^2$ bounded.$ $
My ...
- 1,644
8
votes
4
answers
2k
views
When is the boundary of an open planar set a Jordan curve?
Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve?
Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem.
My ...
- 247
8
votes
1
answer
1k
views
Spectra of a Symmetric Toeplitz Operator
For a physics application, I would like to be able to compute the eigenvalues of the linear operator (acting on the Hilbert space $\ell^2$) given by an infinite matrix of the form
$\begin{bmatrix}
...
- 81
8
votes
2
answers
1k
views
Link between the hairy ball theorem and the fundamental theorem of algebra
I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...
- 81
8
votes
1
answer
656
views
When is the adjoint of a hypoelliptic operator also hypoelliptic?
Suppose that $M$ is a smooth manifold with a measure $\mu$ and let $L^2(M, \mu)$ be a space of all square-integrable functions on $M$.
Recall that $L$ is a hypoelliptic differential operator if for ...
- 329
8
votes
1
answer
1k
views
Beautiful examples of arc-like continua
A continuum is a nonempty compact, connected metric space.
A continuum $X$ is called arc-like if, for every $\varepsilon>0$, there is a continuous and surjective function $f:X\to [0,1]$ such that $...
8
votes
1
answer
434
views
Self-dual finite-dimensional complex normed spaces
Suppose $X$ is a complex normed space of dimension 2 or 3 and $X$ is isometrically isomorphic to its dual. Is $X$ a Hilbert space?
Remarks: There are easy counterexamples in the real case, and in ...
- 11.4k
8
votes
1
answer
3k
views
Homotopy equivalence from contractibility of fiber
Suppose $X$ and $Y$ are two $CW$ complexes and $f:X\rightarrow Y$ is a continuous surjection such that fiber of each point (i.e. $f^{-1}(y)$ for each $y\in Y$) is contractible. Does it implies that $...
- 1,713
8
votes
4
answers
6k
views
Connectedness and the real line
It is fundamental to topology that $\mathbb{R}$ is a connected topological space. However, all the topology books that I have ever looked in give the same proof. (the proof I am thinking of can be ...
- 3,830
8
votes
1
answer
355
views
Proving a certain $ C^{*} $-algebraic inequality
Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality
$$
|\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} \|...
- 1,396
8
votes
1
answer
455
views
Closure of $L(\ell^2,\ell^2)$ in $L(\ell^2,\ell^\infty)$
Let $\ell^2$, $\ell^\infty$ denote the usual sequence spaces and let $L(\ell^2,\ell^2)$ the Banach space of bounded linear operators from $\ell^2$ to $\ell^2$ as well as $L(\ell^2,\ell^\infty)$ the ...
- 261
8
votes
0
answers
6k
views
Convex hulls of compact sets
Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
- 8,512
8
votes
2
answers
1k
views
What is the smallest $C^*$-algebra containing the "standard" pseudodifferential operators?
Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying
$
|\partial^\alpha_x \partial^\beta_\xi a(x,\xi)| \leq C_{\alpha,\beta} (1+|\xi|)^{-|\beta|}
$
) a $C^*$-algebra?
In other ...
- 7,197
8
votes
0
answers
838
views
Intersections of open sets and $\alpha$-favorable spaces
I would like to ask about some classes of topological spaces whether they have been studied, what are they called (if they have a name) and whether they have some interesting properties. For the ...
- 4,713
8
votes
2
answers
577
views
Ultracoproducts and Cartesian products
Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...
- 1,786
8
votes
3
answers
606
views
Compact Hausdorff and C^*-algebra "objects" in a category.
This is yet more on "algebraic objects in functional analysis".
Since Compact Hausdorff spaces are algebraic over Set, it seems to follow that one can find "Compact Hausdorff objects" in any suitable ...
- 26.8k
8
votes
1
answer
708
views
Immersion of $S^1$ in $\mathbb{R}^2$ that can be extended to $\mathbb{D}$
I was wondering about $\mathcal{C}^{\infty}$ immersions $S^1 \longrightarrow \mathbb{R}^2$ which are the restriction to $\partial \mathbb{D}$ of an immersion $\overline{\mathbb{D}} \longrightarrow \...
- 2,696
8
votes
2
answers
579
views
Totally disconnected subspaces
This question is motivated by this one, where no simple solution within ZFC seems to exist. Let me ask a weaker question then.
Suppose that $K$ is a compact, Hausdorff, non-metrizable space. Does it ...
- 81
8
votes
1
answer
1k
views
Is the p-norm of a matrix strictly log-convex?
Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is,
$$
\|A\|_p = \sup_{x\neq 0}\frac{\|Ax\|_p}{\|x\|_p}...
- 3,497
8
votes
2
answers
592
views
Base change for category objects in topological spaces
I was prompted by this question, but the motivation is different.
Suppose we have an internal category object in topological spaces, i.e. an object space X and a morphism space Y, together with ...
- 52.6k
8
votes
3
answers
584
views
"All retracts are closed" as separation axiom
The starting point of this question is the fact that any retract of a $T_2$-space is closed.
Let's say a topological space $(X,\tau)$ is $T_{\textrm{rc}}$ if all retracts of $X$ are closed.
All $T_{\...
- 48.9k
7
votes
1
answer
241
views
On the cardinality of ccc spaces with a $G_\delta$-diagonal
In a recent MO post it was noted that Uspenskij's old example of a Tychonoff ccc space with a $G_\delta$ diagonal and arbitrarily large cardinality is not normal. See:
How could I see quickly that ...
- 4,413
7
votes
2
answers
2k
views
Questions on topologies on space of Radon measures
Consider the space $C_c(\mathbb{R})$ of continuous real-valued functions on $\mathbb{R}$ equipped with the inductive limit topology by $C_c(\mathbb{R}) = \bigcup_{n \in \mathbb{N}} C_c(\mathbb{R}, K_n)...
- 1,773