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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-...
Pedro Lauridsen Ribeiro's user avatar
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 ...
Susp1cious's user avatar
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 ...
Dominic van der Zypen's user avatar
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....
Daniel Spector's user avatar
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 ...
Jason Reed's user avatar
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 \...
yada's user avatar
  • 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 ...
bernhard's user avatar
  • 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 ...
Tom's user avatar
  • 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 ...
asv's user avatar
  • 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 ...
Bazin's user avatar
  • 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 ...
Stefan Waldmann's user avatar
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 ...
Gabriel's user avatar
  • 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 ...
Hannes Thiel's user avatar
  • 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 $...
Julian's user avatar
  • 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 ...
Saal Hardali's user avatar
  • 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 $A, B \in M_N$ ...
Mustafa Said's user avatar
  • 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,...
user avatar
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 $n_i = 1$, ...
Vivek Shende's user avatar
  • 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 ...
Jacob Augstine's user avatar
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 ...
M.González's user avatar
  • 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 ...
TopologicalDynamitard's user avatar
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 ...
ErwinSchr's user avatar
  • 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\...
Arnold Neumaier's user avatar
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\...
Bertrand's user avatar
  • 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 ...
Michael's user avatar
  • 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{...
Juan Bermejo Vega's user avatar
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 ...
Igor Belegradek's user avatar
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 ...
André Henriques's user avatar
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 ...
Matt Rosenzweig's user avatar
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 ...
Leandro's user avatar
  • 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. ...
Julian Hölz's user avatar
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 \...
Ali's user avatar
  • 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 ...
Jochen Glueck's user avatar
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 ...
Zinkin's user avatar
  • 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 ...
Henry.L's user avatar
  • 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 $l^\infty$?
Amir Bahman Nasseri's user avatar
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 ...
Malik Younsi's user avatar
  • 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 ...
Ali's user avatar
  • 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)...
Mizar's user avatar
  • 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 ...
Tanmoy Paul's user avatar
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 ...
Jochen Wengenroth's user avatar
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 ...
Fedor Petrov's user avatar
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 $...
Kurisuto Asutora's user avatar
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{...
tmh's user avatar
  • 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 $A_n:=I_n-M_n^2$. ...
T. Amdeberhan's user avatar
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 (...
user56097's user avatar
  • 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 ...
Goulifet's user avatar
  • 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" ...
Chris Wendl's user avatar

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