All Questions
10,448 questions
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
3
answers
3k
views
Are uniformly continuous functions dense in all continuous functions?
Suppose that $X$ is a metric space. Is the family of all real-valued uniformly continuous functions on $X$ dense in the space of all continuous functions with respect to the topology of uniform ...
- 233
13
votes
5
answers
1k
views
Packing obtuse vectors in $\mathbb{R}^d$
I came across this attractive theorem:
Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that
form an obtuse angle with one another.
This was proved1 as a corollary of a lemma about ...
- 151k
13
votes
6
answers
2k
views
Interesting examples of non-locally compact topological groups
Harmonic analysis is concentrated mostly on studying locally compact groups. I would like to ask people about examples of non-locally compact topological groups that are interesting in connection with ...
13
votes
6
answers
3k
views
Sets with equal positive measure in every interval
Hi,
I want to write a proof that relies on the fact that:
There are Borel Sets $A$ and $B$ contained in $\mathbb{R}$ such that
$A \cap B = \emptyset$ and $\lambda(A \cap (x,y)) = \lambda(B \cap (x,...
- 133
13
votes
2
answers
1k
views
Calkin Algebra and the embedding
Let $H$ be a separable, infinite dimensional Hilbert Space and $Calk(H):=B(H)/K(H)$ denotes
the Calkin algebra. There is obvious surjection $\pi: B(H) \to Calk(H)$ but I'm interested
in somehow ...
- 9,330
13
votes
2
answers
1k
views
Homotopy groups of Fredholm operators
If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that
$$ \pi_0(\mathcal{F}) = \mathbb{Z}\, , $$
i.e. the connected ...
- 173
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
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
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
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
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
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
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
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
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
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
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
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
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
409
views
Does the $\overline{\partial}$ operator have closed image?
Let $X$ be a complex-analytic manifold, not necessarily compact.
Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
- 890
13
votes
2
answers
1k
views
A matrix norm inequality
Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that
$\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...
- 1,748
13
votes
1
answer
4k
views
Modulus of Continuity
I originally posted this question on math.stackexchange (https://math.stackexchange.com/questions/83182/modulus-of-continuity-take-2), but it's been a few days and I haven't received any correct ...
- 29.2k
13
votes
2
answers
3k
views
What is the "correct" generalization of operator norms for nonlinear operators?
I have been recently wondering what is a (or even the) "correct" generalization of the notion of an operator norm to nonlinear operators?
Please excuse the naivete of my question; if you think that ...
- 28.6k
13
votes
1
answer
528
views
Are the logarithms of the integer polynomials discrete in $L^1$ of the unit circle?
Tautologically, the integer polynomials form a discrete set in $L^1$ of the unit circle. On the other hand, a set of logarithms ordered by norm becomes generally rather denser than the original set.
...
- 13.8k
13
votes
2
answers
484
views
$\frac{d}{dt} (A+t B)^p\,\text{ for } p\geq 1$
Given two positive self-adjoint operators $A,B$ on a Hilbert space. Let $p\geq 1$.
I would like to calculate $$\frac{d}{dt}|_{t=0} (A+tB)^p,$$
where the power is defined through the spectral theorem....
- 437
13
votes
2
answers
946
views
Computing a large permanent
Is there a practical way to compute the permanent of a large ($91 \times 91$) $(0,1)$ matrix?
I have tried to use the matlab function written by Luke Winslow which works great for smaller matrices ...
- 6,962
13
votes
1
answer
401
views
Is there a reflexive Banach space whose ball is not the convex hull of its extreme points?
Let $X$ be a reflexive Banach space. Then the convex hull of the extreme points of the unit ball is weakly dense by the Krein-Milman theorem and Kakutani's theorem. My question is, if there is an ...
- 633
13
votes
1
answer
461
views
Does locally nilpotent imply nilpotent for continuous self-maps of intervals?
Let $f\in C([0,1],[0,1])$ be such that:
$$\forall x\in [0,1], \; \exists k\in \mathbb N, \; f^{\circ k}(x)=0.$$
Is it true that $f$ is nilpotent (i.e., that there is some $k$ such that $f^{\circ k}=0$)...
- 4,074
13
votes
1
answer
465
views
One question about the $\eta$ invariant
This question is from the paper, The Analysis of Elliptic Families
II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8.
Suppose ...
- 1,915
13
votes
1
answer
675
views
Wavelet-like Schauder basis for standard spaces of test functions?
Edit: A more precise formulation of my question follows the separation line.
The Schwartz space of test functions $\mathcal{S}(\mathbb{R})$ is isomorphic to $\mathfrak{s}$ the space of sequences of ...
13
votes
1
answer
810
views
Inner and extendible automorphisms of C*-algebras
If an automorphism $\alpha$ of a C*-algebra $A$ is inner then whenever $A$ is a subalgebra of another C*-algebra $B$, $\alpha$ obviously extends to $B$.
Is the converse true: if an automorphism $\...
- 1,786
13
votes
2
answers
610
views
Is there a Gelfand-Naimark-like characterization of group algebras $L_1(G)$?
The Gelfand-Naimark theorem establishes that a complex commutative Banach algebra $A$ with an identity and an involution $x\to x^*$ satisfying $\|x x^*\|=\|x\|^2$ is (isometrically isomorphic to) a $C(...
- 4,461
13
votes
1
answer
724
views
Trace-class operator satisfies $\sum |\lambda_n|<\infty$?
Here's an "exercise" which I thought should be easy, but which I find myself unable to do.
Let $V$ be a Banach space.
Recall that an operator $f:V\to V$ is trace-class if it is in the image of the ...
- 43.2k
13
votes
3
answers
710
views
Completeness of nonharmonic Fourier Series
I have the following question:
The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$.
Thus, certainly the oversampled system $\Phi:...
- 131
13
votes
1
answer
729
views
Making sense of the formula $\operatorname{Det} (I+M )= e^{\operatorname{Tr} \ln (I+M)}$, especially in the infinite dimensional cases
$\DeclareMathOperator\Det{Det}\DeclareMathOperator\Tr{Tr}$In physics literature dealing with quantum field theory, the formula
\begin{equation}
\Det(I+M) = e^{\Tr \ln(I+M)}
\end{equation}
appears ...
- 3,477
13
votes
1
answer
347
views
Existence of a translation-invariant basis of $\ell^2$
This question is heavily inspired by this other one, but is meant to be a hopefully more accessible variant of it (and I think slightly more natural).
I give four equivalent formulations of the same ...
- 32.5k
13
votes
2
answers
656
views
Random matrix with given singular values
Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let
$$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\...
- 1,495
13
votes
2
answers
696
views
C$^*$-algebras isomorphic after tensoring
From the negative answer to this question we know that C$^*$-algebras that are isomorphic after tensoring with $M_n$ for all $n\geq 2$ need not be isomorphic. So what happens when we strengthen this?
...
- 3,984