All Questions
10,241 questions
7
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556
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Non continuous Linear form on $E=C([0,1],\mathbb{R})$ without AC
Let's note $E=C([0,1],\mathbb{R})$ the Banach space of real continuous funtions from the [0,1] interval with the uniform norm.
Is it possible to show a non-continuous linear form on $E$ exists ...
- 1,355
7
votes
4
answers
1k
views
Radon-Nikodým property of $\ell^\infty$
I am wondering whether $\ell^\infty(\mathbb N)$ has the Radon-Nikodým property. Of course $\ell^1(\mathbb N)$ does, but I was unable to find out whether (e.g.) duals of spaces with the R-N property ...
- 3,388
7
votes
1
answer
330
views
Is this a characterization of commutative $C^{*}$ algebras?
Assume that $A$ is a $C^{*}$ algebra with self adjoint elements $A_{sa}$. Assume that for all $a,b\in A$ we have $$ab\in A_{sa} \iff ba \in A_{sa}$$
Is $A$ necessarily a commutative ...
- 356
7
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4
answers
1k
views
Is the space of tempered distribution second countable?
Let $\mathcal S '(\mathbb R^d)$ be the space of Schwartz tempered distributions equipped with the weak-* topology. I need to know if this space is second countable, i.e. if this topology has a ...
- 630
7
votes
2
answers
2k
views
Characterization of Schur's property
A Banach space $H$ is said to have Schur's property if weak convergence of a sequence implies converge in norm. The most famous example of such a space is $\ell^1(\mathbb N)$, while $L^1[0,1]$ does ...
- 2,933
7
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3
answers
4k
views
Is a semicontinuous real function Borel measurable?
Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...
- 1,399
7
votes
3
answers
1k
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Reference on semigroup theory and parabolic PDEs
Recently started to study semigroup theory. My background is equivalent to the first three chapters of the Jack Hale's book "Asymptotic behavior of dissipative systems".
Looking for a reference to an ...
user11178
7
votes
3
answers
4k
views
infinitely many linear equations in infinitely many variables
Let $(a_{mn})_{m,n\in\mathbb{N}}$ and $(b_m)$ be sequences of complex numbers.We say that $(a_{mn})$ and $(b_m)$ constitute an infinite system of linear equations in infinitely many variables if we ...
7
votes
3
answers
909
views
Using the Stone-Weierstrass theorem to solve an integral limit
The following question was posted on math stack exchange here but it got no answers
Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
- 331
7
votes
1
answer
1k
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Cartesian product of Banach spaces: all norms such that the inclusion is an isometry are equivalent?
Let $\mathcal{A}$ be an arbitrary (typically infinite-dimensional) Banach space with norm $\|\cdot\|_{\mathcal{A}}$ and let $\mathcal{A}^{n}$ be its Cartesian product. I came across the following ...
- 141
7
votes
2
answers
319
views
Resource request: Function spaces properties
Is there any resource (book, article,...) that presents all the basic function spaces (Metrizable, Normed, Banach, Sobolev, Hilbert, $L^p$, C[a,b], etc.) together with their properties (completeness, ...
- 1,759
7
votes
3
answers
1k
views
Gamma-function analogues for Gauss sums
I have a Gauss sum, which I have to calculate. I have heard that it has an analogues form with the Gamma function, but couldn't find its formula shape. It would be so nice of you to help me and write ...
- 71
7
votes
4
answers
973
views
I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?
The set of all smooth maps $S^1\to M^n$ ($M$ is a smooth manifold) is a generalized manifold(see http://ncatlab.org/nlab/show/smooth+loop+space).
I was wondering if the set of singular loops (maps ...
- 5,063
7
votes
2
answers
841
views
Why is $\mathbb R^{\mathbb N}$ not high-dimensional enough?
In this paper [1], the authors consider the limiting distribution of $$S_{n,p}:=\frac{1}{\sqrt n}\sum_{k=1}^nX_k$$ for $p\rightarrow\infty$ as $n\rightarrow\infty$, where $X_1, X_2,\dots, X_n$ are ...
- 181
7
votes
2
answers
983
views
English translation of Hilbert's work
Does anyone know if there is an English translation of Hilbert's:
"Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, Leipzig, 1912". ??
Thanks,
Andre
- 111
7
votes
2
answers
2k
views
Does anyone know what is the right reference for the following simple lemma from harmonic analysis?
The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds
$$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(...
- 1,881
7
votes
2
answers
1k
views
A characterization of Hilbert spaces?
My question was prompted by an earlier MO by @Daniel:
Duality map in strictly convex Banach spaces
I will even use his symbol $\phi$ below.
Let $B$ be an ...
- 7,500
7
votes
2
answers
921
views
Exotic spectrum of Laplace operator
Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator,
it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
- 101
7
votes
2
answers
3k
views
Distributions and measures
Hello,
After reading the previous post, I still have some doubts. Let's consider everything on $R$ to avoid complications.
Can we say that any distribution $\mu\in\mathcal{D}'(R)$ of zero order is ...
- 1,649
7
votes
3
answers
814
views
Preduals of B(E)
For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators.
If $E$ is a Banach space then is it known whether
$B(E)$ is always a ...
- 1,411
7
votes
2
answers
464
views
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
- 12.6k
7
votes
1
answer
737
views
Converse of closed graph theorem
Suppose $X$ is a normed linear space. If for every Banach space $Y$ and for every linear operator $T:X\to Y$, graph of $T$ is closed implies $T$ is continuous, then can we prove that $X$ is a Banach ...
- 585
7
votes
2
answers
276
views
Completeness of coefficient functionnals
My questions is about Schauder bases and more specifically about coefficient functionals.
Let $(x_n)$ be a Schauder basis of a Banach space $X$. Thus for all $x$ in $X$, $x = \sum f_n(x) x_n$. The $...
- 183
7
votes
3
answers
754
views
Duality between Banach spaces and compact convex spaces
I always had the impression that there was a duality (i.e. a contravariant equivalence of categories) between Banach spaces and certain notion of pointed compact convex set (something like algebras ...
- 42.4k
7
votes
3
answers
2k
views
$C^1$-functions on Banach spaces
For Banach spaces $X,Y$ and an open subset $U$ of $X$ a function $f:U\to Y$ is $C^1$ if $U\to L(X,Y)$, $x\to f'(x)$ is continuous where, by definition, the derivative $f'(x)$ is a continuous linear ...
- 16.5k
7
votes
1
answer
775
views
Is there a generalized Birkhoff ergodic theorem?
Is there a Birkhoff ergodic theorem for two measure preserving transformations $T$ and $S$ where $S\circ T= T \circ S$ so that $\frac{1}{n+1}\frac{1}{m+1}\sum_{i=0}^{n}\sum_{j=0}^{m}f \circ T^{i}\circ ...
- 71
7
votes
2
answers
632
views
Inverse of partial differential operator as a smooth tame map
Tameness for maps is one of the main ingredients for the Nash-Moser inverse function theorem. A linear map $f: X \to Y$ between Fŕechet spaces with fixed seminorms is called tame if we have an ...
- 5,824
7
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is $L^2(H)$ separable?
Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Is the Hilbert space $L^2(H,\gamma)$ separable?
- 8,512
7
votes
2
answers
1k
views
Energy levels of double well potential
Consider the (quantum) Hamiltonian on the real line
$$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$
Let us assume that the potential $V$ is an even smooth functions with exactly two non-degenerate ...
- 21.8k
7
votes
1
answer
547
views
Is this operator bounded?
Let $T$ be an invertible positive operator and $S$ be another positive operator on a complex Hilbert space.
We then study
$$ \Vert (T+S)^{-1/2}T(T+S)^{-1/2}\Vert$$
I would assume that this norm is ...
- 192
7
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1
answer
771
views
Famous but unavailable paper of Jan Boman
The following paper is well known, but hard to find:
J. Boman, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982.
In this paper ...
- 28k
7
votes
2
answers
837
views
measurable linear functionals are also continuous on separable Banach spaces?
It is well known continuous linear functionals are (Borel) measurable. I have read, as a remark, the converse is also true for separable Banach spaces, but I could not find any references.
- 319
7
votes
2
answers
689
views
Which C*-algebras are complemented in their bidual?
Every von Neumann algebra is 1-complemented in its bidual, and so is every injective C*-algebra. Also, if $C_0(X)$ is infinite-dimensional and separable then it is not complemented in its bidual, and $...
- 3,816
7
votes
4
answers
5k
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Intuitive functional analysis book
I want to know functional analysis book like Terence tao's real analysis and measure theory book, full of intuition. I am aware of linear algebra, real analysis, measure theory, Probability theory.
7
votes
2
answers
1k
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C*-algebraic representation of observables vs self-adjoint operators one
I am trying to reconcile the "physicist" definition of an observable: self-adjoint operator on a Hilbert space, and the operational one as given by Strocchi in "An introduction to the mathematical ...
- 585
7
votes
1
answer
848
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Caccioppoli-Leray Inequality for De Giorgi's theorem proof
I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients.
This is the translation of the original paper
De Giorgi paper
At page ...
- 265
7
votes
3
answers
1k
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spectacular applications of functional analysis in resolutions of apparently unrelated problems
What are some of the spectacular applications of functional analysis to apparently unrelated problems.One that
comes to my mind is Per Enflo's resolution of Hilbert's 5th problem.There are also ...
7
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3
answers
1k
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Non-Borel subspace of Banach space
Let $X$ be a separable Banach space, $M \subset X$ a linear subspace. Must $M$ be a Borel set in $X$?
I believe the answer is "no," since I have seen authors who are careful to talk about "Borel ...
- 29.8k
7
votes
1
answer
453
views
Strong limits of nilpotent operators
Let $H$ be an infinite-dimensional Hilbert space.
Is it possible that the Identity $H\to H$ is a strong limit of nilpotent compact operators?
user473423
7
votes
2
answers
398
views
Quantifying difficulty of integrals versus inverses
Recently, I have been discussing inverses with a tenth grade class and integrals with an eleventh/twelfth grade class, and this has led me to the following wonder:
Wonder. Is there a "reasonable&...
- 7,831
7
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1
answer
562
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On the geometric Hahn-Banach theorem
Let $ X \subset \ R ^ n $ be a closed convex set and
let $ L $ be a straight line such that $ X \cap L = \emptyset $.
Does there exist a hyperplane containing
$ L $ that does not intersect $ X $ ?
...
- 1,503
7
votes
1
answer
856
views
Compactness of set of indicator functions
Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set
$$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$
Is this set compact in $L^\infty(0,1)$ with respect ...
- 395
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2
answers
508
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Making the Fourier transform quantitative
I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website.
I understand ...
- 225
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2
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573
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Existence of spectral gap
I would like to start by saying that any comment or idea is highly appreciated.
Let us observe that for Hilbert-Schmidt operators $H_1,H_2$ on an infinite-dimensional separable complex Hilbert space $...
- 95
7
votes
3
answers
1k
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Simultaneous diagonalization of self-adjoint operators on Hilbert space
Suppose I'm given a finite set of possibly unbounded commuting self-adjoint operators $T_i : \mathfrak H \supset \mathscr D(T_i)\to \mathfrak H, i = 1 , \dots , N$ on a Hilbert space (in the sense of ...
- 203
7
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1
answer
849
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Hahn-Banach theorem for arbitrary locally compact fields?
Does anyone know if the Hahn-Banach theorem is true for every locally
compact field? Specifically, let $F$ be a finite algebraic extension of
either $Q_p$, the $p$-adic completion of $Q$, or of
$S_p$,...
- 383
7
votes
2
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2k
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Arbitrary union of meagre open sets
Let $X$ be a topological space. A subset $M$ of $X$ is called meagre (or of first category) if it is covered by the union of a countable family of closed subsets of $X$ with empty interior.
Can you ...
- 73
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3
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2k
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Why do we distinguish the continuous spectrum and the residual spectrum?
As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension.
If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ ...
- 391
7
votes
2
answers
622
views
Is there a tropical analogue of a reproducing kernel Hilbert space?
In classical functional analysis, one can construct a reproducing kernel Hilbert space by starting with a positive definite kernel, say
$K: [0,1]\times [0,1] \rightarrow \mathbb{R}$.
One then creates ...
- 1,666
7
votes
3
answers
498
views
Sums of unitaries with small norm in full group $C^*$-algebras
Suppose $G$ is a finitely generated group, with given generating set $S={g_1, \dots, g_n}$. (Assume that if $g\in S$, then $g^{-1}\notin S$. (EDIT: Also assume that $S$ is minimal in the sense that ...
- 2,361