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14 votes
6 answers
3k views

What's a natural candidate for an analytic function that interpolates the tower function?

I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that ...
John Jiang's user avatar
  • 4,466
14 votes
4 answers
550 views

About the existence of characters on $B(X)$

Let $X$ be a Banach space. Let $B(X)$ be the space of all bounded linear operators on $X$. Does $B(X)$ have an empty character space for any $X$? I know the proof of the fact that $M_n(\mathbb{C})$ ...
User93709's user avatar
  • 355
14 votes
5 answers
4k views

Is there an extension of the Arzela-Ascoli theorem to spaces of discontinuous functions?

The Arzela-Ascoli function basically says that a set of real-valued continuous functions on a compact domain is precompact under the uniform norm if and only if the family is pointwise bounded and ...
weakstar's user avatar
  • 943
14 votes
3 answers
1k views

Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?

Recall that we say that a bounded measurable set $S\subset\mathbb R^n$ is said to be Caccioppoli if the indicator function $1_S$ is BV, and we set $$ \operatorname{perim}(S)=\| \nabla 1_S\|_{TV} $$ ...
Dominic Wynter's user avatar
14 votes
2 answers
892 views

Do distance functionals separate probability measures?

Let $(\Omega,d)$ be a compact metric space and $\mathcal P(\Omega)$ its space of Borel probability measures. Let $D=\{ d_p\mid p\in\Omega\}$ where $d_p(x)=d(p,x)$ be the set of all "distance ...
Christian Bueno's user avatar
14 votes
2 answers
6k views

Are weak and strong convergence of sequences not equivalent?

For some infinite-dimensional Banach spaces $E$, it is easy to find sequences $\langle x_i:i\in\mathbb N_0\rangle$ which converge to zero weakly but not in the norm topology, i.e. we have $\lim_{i\to\...
TaQ's user avatar
  • 3,584
14 votes
2 answers
1k views

Are smooth functions tame?

I know the article of Hamilton on the inverse function theorem of Nash and Moser (with the same title) where he proves that $C^\infty(M)$ is a tame Fréchet space, when $M$ is closed or compact with ...
Matthias Ludewig's user avatar
14 votes
4 answers
3k views

Representing a product of matrix exponentials as the exponential of a sum

In Proof of a conjectured exponential formula, R. C. Thompson (1986) [edit: apparently, assuming Horn's conjecture] proved that if $A$ and $B$ are Hermitian matrices, then there exist unitary matrices ...
Suvrit's user avatar
  • 28.6k
14 votes
2 answers
2k views

Is the composition of two nowhere differentiable functions still nowhere differentiable?

Let $f,g:\mathbb R\to\mathbb R$ be two continuous but nowhere differentiable functions. By the Denjoy–Young–Saks theorem for almost every point $x_0\in\mathbb R$ one has $$ \limsup\limits_{x\to x_0}\...
Liding Yao's user avatar
14 votes
3 answers
3k views

The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$

Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group ...
yeshengkui's user avatar
  • 1,373
14 votes
1 answer
1k views

Stone-Weierstrass Theorem without AC

To what extent does the usual Stone-Weierstrass Theorem depend on some form of the Axiom of Choice? There seems to be a lot of literature on constructive versions in toposes, but I have been unable ...
Bruce Blackadar's user avatar
14 votes
1 answer
830 views

Spectrum of matrix involving quantum harmonic oscillator

The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator $$a ^* = x- \partial_x \text{ and }a = x+\partial_x.$$ Fix two numbers $\alpha,\beta \...
Kung Yao's user avatar
  • 192
14 votes
1 answer
694 views

Criterion for a Banach algebra to be finite dimensional

Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional. Question. Does it follow that $A$ is finite dimensional? ...
Jochen Glueck's user avatar
14 votes
4 answers
1k views

Is every continuous microlocal operator a pseudo-differential operator?

Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space. Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal. By being microlocal I mean that the wave ...
Joonas Ilmavirta's user avatar
14 votes
1 answer
2k views

Infinite tensor product of states

Tensor products of finite number of different objects are always well described in the literature. However, the situation of infinite tensor products seems to be much tougher. Even in the simplest ...
Glacier's user avatar
  • 143
14 votes
4 answers
1k views

$L^p$ norm means

Consider the unit sphere $S_p^{n-1}$ of an $L^p$ normin $\mathbb{R}^n.$ The question is: what is the expected value of the $L^q$ norm on $S_p^{n-1}?$ Since (I assume) this is intractable in closed ...
Igor Rivin's user avatar
  • 96.4k
14 votes
2 answers
2k views

Is this property equivalent to Lusin's property (N) for continuous functions?

A function $F:[0,1]\rightarrow\mathbb{R}$ satisfies Lusin's (N) property if for every measure zero set $A\subseteq [0,1]$, $F(A)$ has measure zero. (This includes the assertion that $F(A)$ is ...
Linda Brown Westrick's user avatar
14 votes
2 answers
926 views

"Explicit" embedding of $\ell^1$ as a closed subalgebra of a direct sum of matrix algebras

For sake of brevity let $A$ denote the Banach algebra formed by equipping $\ell^1({\mathbb N})$ with pointwise multiplication. This algebra is clearly not isomorphic as a Banach algebra to any uniform ...
Yemon Choi's user avatar
  • 25.8k
14 votes
1 answer
668 views

Why are we interested in spectral gaps for Laplacian operators

Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a spectral gap for such a $\Delta$, that is, finding an interval $(0,c)$ which does not ...
Sven Mortenson's user avatar
14 votes
2 answers
873 views

Which finite dimensional Banach spaces can be represented isometrically as spaces of bounded operators on a finite dimensional Hilbert space?

Background: It is known that every Banach space $X$ can be embedded isometrically as a subspace in the space $C(K)$ of continuous functions on a compact Hausdorff space $K$. Indeed, one can take $K$ ...
Orr Shalit's user avatar
14 votes
1 answer
922 views

What are the applications of the Mazur-Ulam Theorem?

Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ...
Pietro Majer's user avatar
  • 60.5k
14 votes
2 answers
588 views

Heuristic interpretation of the 'third index' for Besov and Triebel-Lizorkin spaces

For $p,q \in (0,\infty)$ and $s \in \mathbb{R}$, one can define certain function spaces, $B_s^{p,q}(\mathbb{R}^n)$ and $F_s^{p,q}(\mathbb{R}^n)$, the Besov and Triebel-Lizorkin spaces respectively. ...
user avatar
14 votes
2 answers
1k views

Order-preserving operator norms

Let us regard the $n\times n$ matrices as operators on the $n$-dimensional $\ell_p$ space; that is, we consider them as linear operators $\ell_p^n\to \ell_p^n$. When $p=2$, $M_n$ is a C*-algebra and ...
Tomasz Kania's user avatar
  • 11.3k
14 votes
2 answers
536 views

Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...
Tobias Diez's user avatar
  • 5,824
14 votes
3 answers
768 views

Is $C^{\infty}(\mathbb{R}^{m+n})$ a flat module over $C^{\infty}(\mathbb{R}^{m})$?

For $m>0$ we consider the ring $C^{\infty}(\mathbb{R}^{m})$ of smooth functions on $\mathbb{R}^{m}$. For $n>0$ we consider the projection $\mathbb{R}^{m+n}\to \mathbb{R}^{m}$ hence $C^{\infty}(\...
Zhaoting Wei's user avatar
  • 9,019
14 votes
2 answers
4k views

What is a good reference that compact resolvent implies Fredholm operator?

Suppose $A \in \mathcal{L}(E_1, E_0)$ is a bounded linear operator between Banach spaces $E_1$ and $E_0$, and we also have that $E_1$ is densely, continuously embedded in $E_0$ (i.e. $A$ can be ...
Jeremy LeCrone's user avatar
14 votes
1 answer
1k views

Any further applications of Freudenthal's 1936 Spectral Theorem?

Seemingly completely forgotten, back in 1936, the Dutch mathematician Freudenthal, quite well known at the time, proved his so called Spectral Theorem, see chapter 6 in Luxemburg & Zaanen : Riesz ...
Elemer E Rosinger's user avatar
14 votes
1 answer
514 views

Generalizing the Fourier isomorphism between Sobolev spaces and weighted $L^2$ spaces to (locally) compact groups?

Motivating examples: Let $V$ be a real vector space with Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$ The ...
Saal Hardali's user avatar
  • 7,789
14 votes
2 answers
2k views

Is the conditional expectation a contraction in weak $\mathbb L^p$ spaces?

Let $(\Omega,\mathcal F,\mu)$ be a probability space. It is well-known that if $\mathcal A$ is a sub-$\sigma$-algebra of $\mathcal F$, $p\geqslant 1$ and $X$ is an element of $\mathbb L^p$ which takes ...
Davide Giraudo's user avatar
14 votes
0 answers
860 views

strong topologies on $C_c^\infty$

UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
Chris Wendl's user avatar
14 votes
0 answers
718 views

Lower bounds on analytic functions connected to Fox H

The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
Tanya Vladi's user avatar
14 votes
0 answers
205 views

Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?

A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article: W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$. In Convex ...
Yemon Choi's user avatar
  • 25.8k
14 votes
0 answers
633 views

Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
Gil Kalai's user avatar
  • 24.7k
14 votes
0 answers
3k views

Tanh version of a Fourier Transform?

I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...
Bill Bradley's user avatar
  • 3,979
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), ...
Ulrich Pennig's user avatar
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 ...
user124775's user avatar
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,...
spoon47's user avatar
  • 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 ...
truebaran's user avatar
  • 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 ...
Chandler's user avatar
  • 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 $B(0,1)$ ...
RadonNikodym's user avatar
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'...
Juan OS's user avatar
  • 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 \...
Peluso's user avatar
  • 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, ...
Andrew Stacey's user avatar
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 ...
erz's user avatar
  • 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)...
Tobias Diez's user avatar
  • 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 ...
Sergei Akbarov's user avatar
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.
user62498's user avatar
  • 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 ...
Matthew Daws's user avatar
  • 18.7k
12 votes
2 answers
3k views

Does there exist an isometry between $L^p$ and $l^p$?

The motivation is simple, as it is trivially right when $p=2$. When considering the duality between $L^p$ ($l^p$) and $L^q$ ($l^q$) when $p$ and $q$ are conjugate in the sense that $1/p+1/q=1$, I ...
S. Li's user avatar
  • 619

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