All Questions
1,150 questions
32
votes
2
answers
4k
views
Are there non-reflexive vector spaces isomorphic to their bi-dual?
Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{**}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.
Are ...
- 7,902
32
votes
19
answers
23k
views
Good books on theory of distributions
Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.
30
votes
8
answers
4k
views
Applications of microlocal analysis?
What examples are there of striking applications of the ideas of microlocal analysis?
Ideally I'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/...
- 7,789
28
votes
7
answers
13k
views
Regular borel measures on metric spaces
When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the ...
- 18.7k
26
votes
3
answers
2k
views
About the category of von neumann algebras
I am looking for one (or more) reference about properties of the category of von Neumann algebra.
More precisely, in an answer of a previous question, Dmitri Pavlov mentions
that the $W^*$ category ...
- 357
26
votes
2
answers
5k
views
Does Arzelà-Ascoli require choice?
Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...
- 29.8k
26
votes
3
answers
2k
views
Universality of zeta- and L-functions
Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
- 7,127
25
votes
2
answers
2k
views
$f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$
Let $f$ be a real function with domain R.
If $f^2$ and $f^3$ are both infinitely differentiable on R,
how to prove $f$ is infinitely differentiable on R?
I have been thinking about this problem for a ...
- 295
25
votes
6
answers
3k
views
Quantum fields and infinite tensor products
As I understand it, a naive interpretation of the state space of a quantum field theory is an infinite tensor product
$$\otimes_{x\in M} H_x,$$
where $x$ runs over the points of space. This ...
- 13.6k
24
votes
2
answers
2k
views
Is the Invariant Subspace Problem arithmetic?
Invariant Subspace Conjecture: A bounded operator on a separable Hilbert space has a non-trivial closed invariant subspace.
Can this conjecture be reformulated as an arithmetic statement, that is, $\...
- 6,891
23
votes
9
answers
2k
views
Nonseparable counterexamples in analysis
When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...
23
votes
4
answers
2k
views
Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?
I consider on $M_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|_2 = \sqrt{\mathrm{Tr}(A^* A)/n}$.
My question is whether a $k$-uple of hermitian matrices that are almost ...
- 9,486
21
votes
2
answers
1k
views
Meager subspaces of a Banach space and weak-* convergence
I previously asked a version of this question on Math.SE, but didn't receive an answer. (But there is a bounty there if you want to claim it!)
Let $X$ be a Banach space. (If it helps, feel free to ...
- 29.8k
21
votes
5
answers
18k
views
When is Sobolev space a subset of the continuous functions?
If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L_2(\Omega): \frac{\partial w}{\partial x_i}\in L_2(\Omega), i=1,...,d)$. My tutor has repeated several ...
- 3,217
21
votes
1
answer
3k
views
Density of polynomials in $C^k(\overline\Omega)$
Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, ...
- 4,034
21
votes
0
answers
732
views
Closed connected additive subgroups of the Hilbert space
It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...
- 60.5k
19
votes
1
answer
3k
views
Infinite convex combinations in a Banach space
Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds:
For any sequence $(x_k)_{k\ge0}$ in $C$, and for
any sequence of non-negative real numbers $(\...
- 60.5k
18
votes
1
answer
5k
views
Unbounded linear operator defined on $l^2$
Let $l^2$ be a Hilbert space of infinite sequences $(z_0, z_1, \cdots)$ with finite $\sum_{i=0}^{\infty} |z_i|^2$.
Are there any simple example of unbounded linear opearator $T: l^2 \to l^2$ with $D(...
- 2,821
18
votes
3
answers
2k
views
Poincare lemma for non-smooth differentiable forms
The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for $C^k$-...
- 16.4k
18
votes
4
answers
1k
views
Who first used the multiplication operator version of spectral theory
This is another history question.
Hilbert phrased the spectral theorem in terms of resolutions of the identity.
While this remained the form of Stone and von Neumann, they did also have the ...
- 827
17
votes
5
answers
3k
views
Conditional probabilities are measurable functions - when are they continuous?
Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
- 8,512
17
votes
3
answers
905
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
- 2,071
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
- 4,952
17
votes
3
answers
3k
views
Why is multiplication on the space of smooth functions with compact support continuous?
I asked the question
Why is multiplication on the space of smooth functions with compact support continuous? on M.SE
sometime ago but I didn't receive a satisfactory answer.
I was reading this ...
- 394
16
votes
2
answers
682
views
Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?
Let $X$ be a Banach space. Consider the map
$$
\alpha\colon X\hat{\otimes} X^* \to B(X)^*,
$$
defined one simple tensors as
$$
\alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, a\...
- 3,497
16
votes
2
answers
731
views
A reference to a characterization of metric spaces admitting an isometric embedding into a Hilbert space
I am looking for a reference to the bipartite version of the Schoenberg's criterion of embeddability into a Hilbert space. The Schoenberg criterion is formulated as Proposition 8.5(ii) of the book &...
- 41.9k
16
votes
2
answers
766
views
Surjectivity of curl
Let: $\mathbb R^3\ni x\mapsto v(x)\in\mathbb R^3$ be a vector field with null divergence belonging to the Schwartz class such that
$$
\int_{\mathbb R^3} v(x) dx=0.
$$
Is it true that there exists a ...
- 16.2k
15
votes
2
answers
3k
views
Generalizations of the Tietze extension theorem (and Lusin's theorem)
I am reasking a year-old math.stackexchange.com question asked by someone else.
(For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.)
The Tietze ...
- 6,287
15
votes
1
answer
2k
views
Bases for spaces of smooth functions
Let $S$ denote the space of rapidly decreasing sequences, which means sequences $a=(a_k)_{k=1}^\infty$ such that the numbers $p_d(a)=\sup\{k^d|a_k| : 1\leq k<\infty\}$ are finite for all $d\in\...
- 56.9k
15
votes
3
answers
2k
views
Can the Riemann integral be defined through a closure/completion process?
Let us consider real-valued functions on the bounded interval $[0,1]$. A "step function" means an element of the vector space spanned by indicator functions of (points and) intervals in $[0,1]$ (the ...
- 32.5k
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\...
- 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 ...
- 9,853
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 ...
- 4,466
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
3
answers
986
views
A conformal map whose Jacobian vanishes at a point is constant?
Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$.
Assume $d \ge 3$ ...
- 6,741
13
votes
0
answers
818
views
Covering number estimates for Hölder balls
Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
- 5,405
13
votes
0
answers
492
views
Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?
Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...
- 82.7k
12
votes
3
answers
1k
views
Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$
Let $X$ be a Banach space. I think that some time ago I read somewhere that, in general, the space $\ell_2(X)$ of all sequences $(x_n)$ in $X$ with $\sum_{n=1}^\infty \|x_n\|^2<\infty$ is not ...
- 4,461
12
votes
3
answers
2k
views
Reference request: Simple facts about vector-valued Sobolev space
Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\...
- 29.8k
12
votes
1
answer
3k
views
Density of smooth functions in Sobolev spaces on manifolds
Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have ...
- 3,383
12
votes
3
answers
2k
views
To what extent is convexity a local property?
A polyhedron is the intersection of a finite collection of halfspaces. These halfspaces are not assumed to be linear, i.e. their bounding hyperplanes are not assumed to contain the origin. The ...
- 1,846
12
votes
1
answer
575
views
Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...
- 4,878
11
votes
3
answers
3k
views
Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix
I am looking for the fundamental solution of the following PDE
$$\partial_i (a^{ij}\partial_j u)=f$$
where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients.
I could find a ...
- 1,303
11
votes
2
answers
714
views
A neat evaluation of an infinite matrix?
Let $M_n$ be an $n\times n$ matrix defined as
$$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
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
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
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 ...
- 711
11
votes
1
answer
258
views
Bilinear product of two summable families
Consider the following statement, which I suspect is false as written:
Let $E,F,G$ be (Hausdorff) topological vector spaces (over $\mathbb{R}$), let $\varphi\colon E\times F\to G$ be continuous and ...
- 32.5k
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
10
votes
1
answer
524
views
Linearization instability and singular points of algebraic varieties
In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...
- 21.5k