All Questions
981 questions
7
votes
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
0
answers
4k
views
Explicit element of $(\ell^{\infty})^* - \ell^1$? [duplicate]
Possible Duplicate:
What’s an example of a space that needs the Hahn-Banach Theorem?
It is well known that the dual of $\ell^{\infty}$ properly contains $\ell^1$ (over $\mathbb{N}$, say). ...
- 25.6k
7
votes
1
answer
703
views
A Question About Pure States, Support Projections and Central Covers
I am trying to study the paper Consistency of a Counterexample to Naimark’s Problem by Charles Akemann and Nik Weaver, and there is a claim in Lemma 1 of the paper that I am stuck at, which is as ...
user36116
7
votes
2
answers
2k
views
Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
- 8,998
7
votes
1
answer
814
views
An equivalent condition for separability of $X^*$
Let $X$ be a Banach space. By the weak operator topology on $B(X)$, we mean the locally convex topology implemented by the following semi-norms:
$$B(X)\to[0,\infty) : T\to|\langle Tx,x^*\rangle|$$
...
- 4,058
7
votes
1
answer
861
views
Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold
I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
- 159
7
votes
2
answers
920
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
Arzelà-Ascoli theorem and Hölder spaces
Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist ...
- 21.8k
7
votes
1
answer
1k
views
Exactness of completed tensor product of nuclear spaces
Let $0 \to V \to W \to L \to 0$ be a strict short exact sequence
of (complete) nuclear spaces, i.e. it is a short exact sequence of
(complete) nuclear spaces, all the maps are continuous, the map $...
- 2,649
7
votes
1
answer
261
views
Comparison of several topologies for probability measures
Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
- 243
7
votes
2
answers
530
views
The kernel of all invariant means
Let $G$ be a discrete group which is amenable (i.e. it admits an left-invariant mean, i.e. a continuous positive normalised linear functional $m:\ell^\infty(G) \to \mathbb{R}$ such that $\forall g \in ...
- 4,432
7
votes
1
answer
210
views
$GL_1(\mathcal{E}'(\mathbb{R}))$ open in $\mathcal{E}'(\mathbb{R})$?
Let $\mathcal{E}'(\mathbb{R})$ be algebra of all compactly supported distributions on $\mathbb{R}$, equipped with the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$, and with the usual ...
- 73
7
votes
1
answer
756
views
Taylor expansion with remainder on locally convex spaces
It is usual to introduce Fréchet and Gâteaux derivatives in Banach spaces. In this context, the familiar Taylor expansion with remainder is also at hand, as you can see on the picture below taken from ...
- 3,535
7
votes
3
answers
6k
views
Integral kernel for the resolvent of the laplace operator
Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of $\Delta$...
- 225
7
votes
2
answers
2k
views
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
7
votes
2
answers
1k
views
Prove that a given distribution is tempered
Suppose I have a distribution $E$ such that $\phi \ast E$ is square-integrable for all $\phi \in C_c^\infty \left( \mathbb{R}^d \right)$. Is it possible to prove that $E$ is tempered? It seems ...
- 651
7
votes
1
answer
439
views
series representation in injective tensor products
All books on tensor products of Banach spaces contain the well-known theorem of Grothendieck that every element of the completed projective tensor product
$X \tilde{\otimes}_ \pi Y$ has a ...
- 16.4k
7
votes
2
answers
684
views
Hölder continuity for operators
Let $x,y$ be positive real numbers then
$$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$
we obtain $1/...
user127612
7
votes
1
answer
219
views
$*$-algebras, completions, and $K$-theory
What is an example of a $*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|_1$ and $\| \cdot \|_2$, with respect to which we can complete $\cal{A}$ to give two $C^*$-algebras $...
- 993
7
votes
1
answer
414
views
Criteria for operators to have infinitely many eigenvalues
Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem.
For non-normal operators this no longer has to be true.
There ...
- 536
7
votes
1
answer
220
views
Is $C^{\infty}(E)$ a projective Frechet $C^{\infty}(M)$-module for a $C^{\infty}$-fiber bundle $E\to M$ with compact fiber?
The question is a special case of a previous question.
Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection ...
- 9,019
7
votes
1
answer
429
views
Open projections and Murray-von Neumann equivalence
Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...
- 619
7
votes
2
answers
665
views
Non-separable metric probability space
Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if:
the support of $\mu$ is contained in a separable subspace of $X$.
Questions:
1. Is there a standard name for this property?
...
- 6,541
7
votes
1
answer
545
views
Is the fundamental group of $II_{1}$ factors invariant under a relation?
In order to define the equivalence relation, let's first recall the Tomita-Takesaki modular theory and conditional expectation for von Neumann algebras.
Let $H$ be a separable Hilbert space and $B(H)$...
7
votes
1
answer
592
views
topologies on U(H)
There are many topologies on the algebra $B(H)$ of bounded operators on Hilbert space:
the weak, strong, ultraweak (also called σ-weak), ultrastrong (also called σ-strong), and some more......
- 43.2k
7
votes
1
answer
572
views
What is $\hat{A}=\{[\pi]:\pi$ is a irreducible representation of $A$} ( $A$ is a $C^*$-algebra)?
Let $A=\{f:[0,1]\to M_2(\mathbb{C}): $f continuous and $ f(0)=\begin{pmatrix} f_{11}(0) & 0 \\ 0 & f_{22}(0) \end{pmatrix}\}$ be a $C^*$-algebra with pointwise multiplication, involutions and ...
- 1,188
7
votes
2
answers
2k
views
Questions on topologies on space of Radon measures
Consider the space $C_c(\mathbb{R})$ of continuous real-valued functions on $\mathbb{R}$ equipped with the inductive limit topology by $C_c(\mathbb{R}) = \bigcup_{n \in \mathbb{N}} C_c(\mathbb{R}, K_n)...
- 1,773
7
votes
2
answers
449
views
Distribution that vanishes against approximated delta is zero
Suppose we have a Schwartz distribution $\phi$ on $\mathbb{R}^d$ such that $$ \forall x, \ \lim_{\lambda \to 0}| \langle\phi, \psi^{\lambda}_x \rangle| =0$$
where $\psi^{\lambda}_{x}=\lambda^{-d}{\...
- 387
7
votes
1
answer
609
views
$H^s$ norm of a solution of a nonlinear Schrödinger equation
I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...
- 71
7
votes
0
answers
304
views
Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories
Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
- 167
7
votes
0
answers
927
views
What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?
I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$ \operatorname{supp} \phi \...
- 1,051
7
votes
1
answer
789
views
Subadditivity of the square root for matrices
For positive numbers $a$ and $b$ we have the inequality $\sqrt{a+b} \leqslant \sqrt{a} + \sqrt{b}$. Is it true that the same holds if we take $a$ and $b$ to be positive semidefinite matrices?
If not, ...
- 5,005
7
votes
0
answers
555
views
maximal tensor product commutes with inductive limits
Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras and let $B$ be an arbitary $C^*$ algebra.
I want to prove $(\varinjlim A_n)\otimes_{max} B \cong \varinjlim (A_n \otimes_{max} B)$. This ...
- 1,188
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
6
votes
0
answers
182
views
Factorization of metric space-valued maps through vector-valued Sobolev spaces
Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that
$$
\int_{x\in X}\,d(y_0,f(x)...
- 5,405
6
votes
3
answers
1k
views
Orthonormal basis in $W^{1,2}([0,1])$
Consider the Hilbertspace $W^{1,2}([0,1])$ (i.e. Sobolev space) with the standard inner product which is defined by: $(f,g) = (f,g)_{L^{2}([0,1])} + (f',g')_{L^{2}([0,1])}$. Here $[0,1]$ is not ...
- 63
6
votes
1
answer
498
views
A finely open set, not open up to polar set?
I already asked this on M.SE, but get no answers.
Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge 2$, which is not ...
- 1,724
6
votes
0
answers
529
views
Infinite-dimensional "algebraic varieties"
This question was also formerly posted on MSE but has not received any answer or comment.
Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its projectivization. ...
- 1,543
6
votes
1
answer
754
views
Banach Manifold
Let $M$ and $N$ be closed manifolds. Is it true that
$C^{k}(N,M)$, which is the space of functions $f: N\to M$ such that $f\in C^{k}$, is a $C^{\infty}$ Banach manifold? If so, can you help me to ...
- 225
6
votes
1
answer
422
views
Absolutely 2-summable operator on a Hilbert space
An bouneded linear operator $A \in L(X, Y)$ (here $X$, $Y$ are Banach spaces) is called absolutely $2$-summable if there exists a $C>0$ such that
$$ \left( \sum_{j=1}^N \| A x_j\|_X^2 \right)^{1/2} ...
- 9,853
6
votes
1
answer
2k
views
Sobolev functions on $\mathbb{R}^N$ cannot be discontinuous on a $(N-1)$-dimensional submanifold
How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?
- 839
6
votes
1
answer
150
views
Examples of non-isomorphic $C^\ast$ algebras with isomorphic quasi-state spaces
Let $A$ (resp. $B$) be a unital $C^\ast$-algebra, $\mathcal{Q}(A)$ (resp. $\mathcal{Q}(B)$) the compact convex subset of $A^\ast$ equipped with the $\sigma(A^\ast, A)$ (resp. $\sigma(B^\ast, B)$) ...
- 1,586
6
votes
3
answers
3k
views
Non-empty resolvent set, then operator closed?
On Hilbert spaces, the following is true:
Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed.
The obvious proof I see to show this uses explicitly the ...
- 115
6
votes
2
answers
2k
views
The functoriality of group C* algebra structure
Let $G$ and $H$ be discrete groups and $f:G \rightarrow H$ be any homomorphism of these groups. I have three questions about it:
1) How to prove the functoriality of the construction of universal $C^*...
- 61
6
votes
1
answer
321
views
Derivatives of norm of vector-valued holomorphic functions
Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think ...
- 5,529
6
votes
0
answers
83
views
Are invertible measures strictly dense?
Let $L_1(\mathbb T)$ be considered as a closed ideal of $M(\mathbb T)$, the Banach algebra of measures on the circle. Then $M(\mathbb T)$ can be identified with the multiplier algebra of $L_1(\mathbb ...
- 113
6
votes
0
answers
169
views
Characterizing fullness of a von Neumann algebra by the topology of its bimodules
Let $\mathcal{M}$ be a $\mathrm{II}_1$ factor. Among other characterizations, it is said to be full iff the adjoint map:
$$
\mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...
- 1,090
6
votes
2
answers
3k
views
Fourier series representing a continuous function?
This is maybe not really research level, but I have not found anything in the literature, and asking on math.stackexchange wasn't successful either.
Fourier series define an isometry $L^2(\mathbb{Z}) ...
- 976
6
votes
1
answer
1k
views
Under what conditions does a continuous linear map map a closed subspace to a closed subspace?
Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$?
It is obviously satisfied if $W$ is ...
- 2,649
6
votes
0
answers
110
views
$C(X)$-Fredholm operators and Atiyah-Jänich theorem
Let $X$ be a compact Hausdorff topological space and consider the Hilbert space $\ell^2(\mathbb N)$. As shown here, any $T\in C(X,\ell^2(\mathbb N))$ induces a $C(X)$-Fredholm operator
$$
\begin{array}...
- 61