All Questions
10,050 questions
11
votes
4
answers
668
views
Is every non-negative test function the limit of a sequence of sums of squares of test functions?
Let $0\leq f\in\mathscr{D}(\mathbb{R}^n)$. As shown e.g. by J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232 (2006) 137-...
- 7,817
11
votes
3
answers
2k
views
How can I simplify this sum any further?
Recently I was playing around with some numbers and I stumbled across the following formal power series:
$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$
I was able ...
- 213
11
votes
1
answer
766
views
Generalized limits on $\ell^\infty(\mathbb{N})$
Let $\ell^\infty(\mathbb{N})$ denote the set of bounded real sequences $(a_n)_{n\in\mathbb{N}}$. The $\lim$ operator is a partial linear operator from $\ell^\infty(\mathbb{N})$ to $\mathbb{R}$. With ...
- 48.9k
11
votes
4
answers
2k
views
problems from the scottish book
Which of the problems from the Scottish Book (pdf of English version) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from ...
11
votes
3
answers
1k
views
Boundedness of the derivative of the trace of an H^1 function
As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical matter....
- 882
11
votes
3
answers
1k
views
Continuous automorphism groups of normed vector spaces?
Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
- 180
11
votes
2
answers
1k
views
Do Hausdorff locally convex inductive limits always exist?
The following is from Schaefer, "Topological Vector Spaces", 1999, p. 56/57:
Let $(E_\alpha)_{\alpha \in A}$ be a family of locally convex spaces with $\alpha$ in a directed poset $A$ and $h_{\beta \...
- 1,773
11
votes
2
answers
1k
views
exp(S) exp(T) = exp(S+T) for commuting operators
The standard way to prove the exponential law for two bounded commuting operators $S, T$
$$
\exp(S)\exp(T) = \exp(S+T)
$$
is to pass by the binomial formula and the power series of $\exp(.)$. I wonder ...
- 111
11
votes
2
answers
1k
views
How to show that something is not completely metrizable
I have a Polish space $X$ and a subset $A \subset X$.
I know that $A$ is completely metrizable (in its induced topology) if and only if $A$ is a $G_\delta$-set in $X$.
This means: If I want to show ...
- 987
11
votes
2
answers
6k
views
Is the $L^1$-space dual to a Banach space
Let $(\Omega,\mu)$ be a measure space. It is well known that for $1<p\leq \infty$ one has the duality
$$L^p=(L^{p*})^*,$$
where $1/p+1/p^*=1$.
Question. Is it known that the Banach space $L^1$ is ...
- 21.8k
11
votes
2
answers
712
views
Poincaré lemma for distributions
Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad ...
- 16.2k
11
votes
4
answers
1k
views
Example of noncomplete quotient of complete lcs mod closed subspace
The following statement is well-known: for a Fréchet space $V$ and a closed subspace $W \subseteq V$ the quotient $V / W$ is again complete and hence a Fréchet space. For the particular case of a ...
- 8,098
11
votes
2
answers
1k
views
Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...
- 721
11
votes
2
answers
638
views
von Neumann algebras as C*-algebras with multiplicative conditional expectation $A^{**}\to A$
Let $A$ be a C*-algebra. We identify $A$ with its canonical image in the bidual $A^{**}$. Consider the following conditions:
(1) $A$ is a von Neumann algebra.
(2) There is a multiplicative ...
- 3,497
11
votes
2
answers
721
views
Existence of an open convex set
Let $T$ be a normed vector space, $K\subseteq T$ compact and convex and $O\subseteq K$ convex and open in $K$ (i.e. open w.r.t. the subspace topology of $K$ inherited by $T$).
Can we find an open set $...
- 113
11
votes
1
answer
668
views
Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?
Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...
- 7,789
11
votes
1
answer
983
views
Applications of the "almost commuting" theorem of H. Lin
H. Lin proved that "almost commuting" hermitian matrices are "nearly commuting." To be more precise, Lin showed that given $\epsilon > 0$ there exists a $\delta > 0$ such that if $A, B \in M_N$ ...
- 3,699
11
votes
1
answer
806
views
Algebraicity of Eigenvectors in a Hilbert space
Let $(e_j)_{j\in\mathbb N}$ be an orthonormal basis of a Hilbert space $V$. Let $T:V\to V$ be continuous, selfadjoint linear operator.
Assume that for all $i,j\in\mathbb N$ the number $\langle Te_i,...
user1688
11
votes
1
answer
602
views
How do analysts think about functions with poles at all roots of unity?
In branches of algebra impinging on the enumeration of partitions, one often encounters formulas like
$$\prod_i \left( \frac{1}{1-q^i} \right)^{n_i}$$
for some integers $n_i$. E.g., with $n_i = 1$, ...
- 8,723
11
votes
3
answers
3k
views
Dual space of $L^2(\mathbb{R},L^1(0,1))$?
I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures)
Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
- 175
11
votes
1
answer
2k
views
Bounded operator on a normed space with empty spectrum
A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space.
Is there any example of bounded operator acting on a ...
- 4,461
11
votes
1
answer
339
views
What is an example of two Banach spaces $X,Y$ such that $X$ embeds isometrically but not linearly into $Y$?
By a result of Godefroy and Kalton if $X,Y$ are separable Banach spaces and $X$ embeds isometrically into $Y$, then $X$ embeds with a linear isometry into $Y$.
Is this result known to fail for ...
11
votes
2
answers
1k
views
Harmonic oscillator in spherical coordinates
It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry.
More precisely, the operator
$$-\frac{d^2}{dx^2}+x^2$$
can be ...
- 113
11
votes
3
answers
661
views
norm inequalities
Let $p>2$. I'd like to know the best possible lower and upper bound for $\|x\|_p$ given that $x\in R^n$ and $\|x\|_1$, $\|x\|_2$, and $\|x\|_\infty$ have fixed values.
It is well-known that
$$\|x\...
- 3,873
11
votes
2
answers
8k
views
About the Fourier transform of the logarithm function
I want to calculate / simplify:
$$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$
where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
- 1,199
11
votes
3
answers
445
views
Does the generator of a 1-parameter group of Banach space isometries know which elements are entire?
Let $X$ be a complex Banach space. Let $(\sigma_t)_{t \in \mathbb{R}}$ be a 1-parameter group of linear isometries of $X$ which is strongly continuous i.e. $t \mapsto \sigma_t(x)$ is continuous for ...
- 662
11
votes
1
answer
2k
views
Understanding Bruhat's notion of Schwartz function
I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwartz-Bruhat space $\mathscr{...
11
votes
2
answers
2k
views
What's wrong with compact-open topology on the space of maps?
Given a smooth vector bundle $E$ with non-compact base, let
$\Gamma(E)$ be the space of $C^\infty$ sections equipped with compact-open $C^\infty$-topology.
I have heard that $\Gamma(E)$ is not ...
- 29.1k
11
votes
1
answer
413
views
Estimating the growth of the Taylor coefficients given the growth of the function at the boundary
Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies
$$
|f(z)|\le \frac{1}{(1-|z|)^{k}}
$$
for some fixed $k>0$.
Question: What can I deduce about the growth of the ...
- 43.2k
11
votes
2
answers
451
views
Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$
I asked this question on Math StackExchange, but it did not receive an answer, despite my offering a bounty to attract attention. I am unsure whether it is appropriate for this venue, but I thought ...
- 873
11
votes
4
answers
2k
views
Is this a $C^{\infty}$ function ?
Let be $(a_n)\in\ell^2(\mathbb N)$ and consider the mapping $f:\ell^2(\mathbb N)\to\ell^2(\mathbb N)$ given by
$$
f\Big((a_n)\Big)=(a_n^n).
$$
Question: Is $f$ a Fréchet $C^{\infty}$ function in whole ...
- 2,044
11
votes
2
answers
932
views
A group action of the Heisenberg group with special symmetries
Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...
11
votes
1
answer
341
views
Density of linear subspaces in $C(K)$
Let $K$ be a compact Hausdorff space and denote by $C(K)$ the space of all real valued and continuous functions on $K$. We endow $C(K)$ with the supremum norm topology, making it a Banach space.
...
- 467
11
votes
1
answer
411
views
A density question for the Hilbert transform
Let $\mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $\mathbb R$. Are the set of functions
$$ \{(f+\mathscr Hf)_{|_{(0,1)}}\,:\, f \in C^{\infty}(\mathbb R)\quad \...
- 4,153
11
votes
2
answers
504
views
On dense embedding of Banach spaces
Disclaimer: When I came up with this question yesterday, I suspected it to be trivial (trivially true or trivially false). Then it kept me awake several hours tonight... (I still hope, though, this is ...
- 12.6k
11
votes
2
answers
1k
views
Concentration compactness. Can this concept be stated in a theorem?
I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk.
When I approached the speaker ...
- 501
11
votes
1
answer
441
views
Example of Banach spaces with non-unique uniform structures
While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform ...
- 8,071
11
votes
1
answer
964
views
Quotients of l^infty
Let $M$ be a closed subspace of $l^\infty$. Suppose that the quotient $l^{\infty}/M$ is isomorphic to $l^\infty$. Is it true that $M$ is complemented in $l^\infty$?
11
votes
1
answer
633
views
Inequivalent complete norms and the axiom of choice
Hi,
I've been wondering about the following :
Is it possible, without the axiom of choice, to have two inequivalent complete norms on a vector space?
All the examples of inequivalent complete norms ...
- 2,154
11
votes
1
answer
582
views
An extension of the Carlson's theorem in complex analysis
For the statement of Carlson's theorem please see,
https://en.wikipedia.org/wiki/Carlson%27s_theorem.
There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...
- 4,153
11
votes
2
answers
551
views
Smoothness of finite-dimensional functional calculus
Assume that $f:\mathbb R\to\mathbb R$ is continuous.
Given a real symmetric matrix $A\in\text{Sym}(n)$, we can define $f(A)$ by applying $f$ to its spectrum. More explicitly,
$$ f(A):=\sum f(\lambda)...
- 3,146
11
votes
1
answer
953
views
Separable bidual but nonseparable third dual
Does there exist a Banach space $X$ such that $X^{**}$ is separable but $X^{***}$ is non-separable?
More generally, for every natural $n$ can someone construct an example of Banach space $X$ such ...
- 521
11
votes
4
answers
2k
views
Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces
This question was posed on MathStackExchange but did not get an answer (even with a bounty).
In all books that I have checked the spectral theorem (every self-adjoint unbounded operator on a Hilbert ...
- 16.4k
11
votes
1
answer
1k
views
Stone-Weierstrass analogue for $L^p$
Let $A$ be a complex algebra of bounded measurable functions on the measure space $(X,\mu)$ (case of $[0,1]$ with Lebesgue measure is enough for me) closed under conjugation. Assume that $A$ separates ...
- 109k
11
votes
2
answers
506
views
Minimization problem for convolution
Let $g(x)$ be a non-negative function supported on $[0,1]$. Let $g \ast g$ denote the convolution of $g$ with itself. Question: What is the smallest possible $L^1(0,1)$ norm of $g$, if I require that $...
- 2,207
11
votes
1
answer
320
views
Conceptual explanation for the appearance of entropy in $\frac{d}{dp}\|x\|_p$
For $x\in \mathbb{R}^d$, an elementary computation yields that
$$\frac{d}{dp}\log \|x\|_p =\frac{1}{p^2}\sum_{i=1}^d \frac{|x_i|^p}{\|x\|_p^p}\log \frac{|x_i|^p}{\|x\|_p^p}=-\frac{1}{p^2}\operatorname{...
- 775
11
votes
2
answers
714
views
A neat evaluation of an infinite matrix?
Let $M_n$ be an $n\times n$ matrix defined as
$$M_n
=\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$
With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. ...
- 43.2k
11
votes
1
answer
302
views
Without AC, which implications between the different definitions of amenability still hold?
More precisely, I would like to know which implications between the following definitions of amenability of a discrete countable (or even finitely generated) group can be proved to hold with only ZF (...
- 402
11
votes
3
answers
4k
views
Structure of an intersection of $L^p$-spaces
In what follows, $L^p$ denotes the space of functions from $\mathbb{R}$ to $\mathbb{R}$ such that $\int_{\mathbb{R}} |f(x)|^p\mathrm{d}x < \infty$.
I am interested to understand the structure we ...
- 2,306
11
votes
1
answer
693
views
discontinuous functions on the Sobolev borderline
The Sobolev embedding theorem implies that every function of class $W^{k,p}$ on a reasonable $n$-dimensional domain is continuous if $kp > n$. Cases with $kp=n$ are known as "borderline" ...
- 486