All Questions
978 questions
2
votes
1
answer
755
views
Existence of a solution to an infinite dimensional Stratonovich SDE
Let
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace
$U_0:=Q^{1/2}U$
$(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$ ...
- 167
2
votes
0
answers
73
views
Alberti rank-one theorem and irregular jump discontinuities
Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (...
- 839
2
votes
1
answer
301
views
Density of continuous functions to interior in set of all continuous functions
Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold with boundary. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed ...
- 5,405
2
votes
1
answer
393
views
Peter-Weyl theorem (compact quantum groups)
I'm reading the paper Notes on compact quantum groups. In this paper, the following theorem is proven:
Question: Why is the marked equality true?
user167952
2
votes
1
answer
586
views
Inverse Problem for Pullback
Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) ...
- 61
2
votes
2
answers
336
views
Metrization of a topological vector space
Let $C(\mathbb R^d)$ be the space of continuous functions on $\mathbb R^d$, and $C_{lip}(\mathbb R^d)\subset C(\mathbb R^d)$ be the subspace of Lipschitz functions. We endow $C_{lip}(\mathbb R^d)$ ...
user128095
2
votes
2
answers
485
views
Dual space of the completion of the space of Lipschitz functions
This question is a continuation of this post : Metrization of a topological vector space
Let $C_{lip}(\mathbb R^d)$ be the space of Lipschitz functions on $\mathbb R^d$. We endow $C_{lip}(\mathbb R^...
user128095
2
votes
1
answer
997
views
Derivative and Jacobian determinant of solution of ODE [closed]
Let $\Phi$ be the unique solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
where we have assumed $f$ smooth.
...
user124345
1
vote
2
answers
535
views
Non-closed range space of Laplace operators?
Set $ -\Delta: H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3) $. Then $ \mathcal{R}(-\Delta) $ is non-closed?
Sorry if this question is trivial. I am not familiar with theory of ...
- 269
1
vote
1
answer
148
views
How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$?
Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$
How can we calculate the generalized gradient $\partial_Cf(x)$ of ...
- 167
1
vote
3
answers
345
views
Under what general conditions is the set $S := \left\{\int_{X}v(x)\pi(x)\,\mathrm{d}P(x) \mid \pi: X \to A\right\}$ closed?
Let $X$ be a compact subset of $\mathbb R^n$ and let $A$ be a compact subset of $\mathbb R^k$. Let $P$ be a probability distribution on $X$ and $v$ be a $P$-measurable function from $X$ to $\mathbb R^{...
- 6,853
1
vote
1
answer
654
views
Properties of the trace term in the Itō formula
Let's consider the SDE $${\rm d}X_t=u_t(X_t){\rm d}t+\xi_t(X_t){\rm d}W_t\;\;\;\text{for all }t\ge 0\tag 1$$ where
$U,H$ are separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ is nonnegative ...
- 167
1
vote
1
answer
247
views
Elliptic interface problem without conditions on the interface
Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...
user60665
1
vote
1
answer
263
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 463
1
vote
1
answer
154
views
BV function with absolutely continuous divergence
Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$.
Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and ...
- 839
1
vote
2
answers
234
views
Find $\inf_{P_{X_1,X_2}}P_{X_1,X_2}(\|X_1-X_2\| > 2\alpha)$ , where $\alpha > 0$ and inf is over couplings
Let $\mathcal X$ be a seperable Banach space with norm $\|\cdot\|$, and let $X_1$ and $X_2$ be random vectors on $\mathcal X$ with finite means.
Question. Given $\alpha > 0$, what is value of, ...
- 6,853
1
vote
0
answers
75
views
Derivation of the vortex filament equation from Euler equation
How can the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
be derived from the Euler equation
$$\partial_t \...
- 277
0
votes
2
answers
494
views
Semifinite measure and spectral theorem
Let $H$ be a complex Hilbert space (not necessary separable).
Spectral Theorem: Let $A_1$ and $A_2$ be two commuting normal operators, then there exists a measure space $(X,\mathcal{E},\mu)$,
two ...
- 1,154
0
votes
2
answers
1k
views
Question on Hartogs's Extension Theorem
Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)?
For Hartogs's Extension Theorem see here:
http://en.wikipedia.org/wiki/...
- 53
0
votes
2
answers
403
views
Application of uniform boundedness principle
$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...
user128095
0
votes
2
answers
125
views
Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 835
94
votes
1
answer
11k
views
The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
- 23.5k
71
votes
2
answers
6k
views
Barrelled, bornological, ultrabornological, semi-reflexive, ... how are these used?
I'm not a functional analyst (though I like to pretend that I am from time to time) but I use it and I think it's a great subject. But whenever I read about locally convex topological vector spaces, ...
- 26.8k
65
votes
9
answers
12k
views
Polish spaces in probability
Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed.
Question: What can go wrong when doing probability on non-Polish spaces?
- 651
59
votes
9
answers
10k
views
Motivation for and history of pseudo-differential operators
Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...
- 5,327
45
votes
7
answers
16k
views
What is an intuitive view of adjoints? (version 2: functional analysis)
After realising that I don't have an intuitive understanding of adjoint functors, I then realised that I don't have an intuitive understanding of adjoint linear transformations!
Again, I can use 'em, ...
- 26.8k
37
votes
2
answers
2k
views
Moving one family of commuting self-adjoint operators to another without losing commutativity on the way
This is actually not a question of mine, so I'll be short on motivation and say nothing beyond that if this were true, a few fancy harmonic analysis techniques that a colleague of mine used in proving ...
- 61.9k
34
votes
2
answers
3k
views
Can we recover a von Neumann algebra from its predual?
By definition, a von Neumann algebra is a C*‑algebra A
that admits a predual, i.e., a Banach space Z such that
Z* is isomorphic to the underlying Banach space of A.
(We require that isomorphisms in ...
- 37.8k
33
votes
1
answer
2k
views
For which maps $S^1\to S^1$ is the winding number defined?
There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• Continuous maps:
Using the unique path lifting property of the universal covering map $\mathbb R\to S^...
- 43.2k
33
votes
1
answer
2k
views
Stone-Weierstrass theorem for holomorphic functions?
The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called
Naсhbin's theorem: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth ...
- 7,426
31
votes
1
answer
2k
views
Topology on space of hyperfunctions
This is a reference request, coming from someone with little knowledge of hyperfunctions:
Which methods have been used to endow the space of hyperfunctions $\mathcal B(\mathbb R)$ with something like ...
- 21.3k
31
votes
3
answers
5k
views
When is an integral transform trace class?
Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator
$$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$
the operator $K$ is Hilbert ...
- 11.2k
30
votes
3
answers
3k
views
Surjectivity of operators on $\ell^\infty$
Can anyone give me an example of an bounded and linear operator $T:\ell^\infty\to \ell^\infty$ (the space of bounded sequences with the usual sup-norm), such that T has dense range, but is not ...
- 301
29
votes
6
answers
9k
views
Nonseparable Hilbert spaces
Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...
- 9,330
29
votes
1
answer
4k
views
Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle
Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...
- 25.5k
28
votes
2
answers
1k
views
Can an operator have Exp(z) as its characteristic "polynomial"?
Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator. Define
$$ f_T(z) = \sum_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$
the ...
- 5,898
27
votes
1
answer
1k
views
The dual of $\mathrm{BV}$
$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
- 683
27
votes
0
answers
1k
views
Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 356
27
votes
2
answers
5k
views
What can be said about the Fourier transforms of characteristic functions?
What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular,
What properties are common to ...
- 2,719
26
votes
2
answers
2k
views
When is a locally convex topological vector space normal or paracompact?
All locally convex topological vector spaces (LCTVS) are completely regular, since their topology is given by a family of semi-norms. I'm interested in conditions that imply that a LCTVS is ...
- 26.8k
26
votes
3
answers
16k
views
the dual space of C(X) (X is noncompact metric space)
It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f:
X\rightarrow \mathbb{C} \text{ is continuous and bounded} \}$ is $M(X)$, the space of ...
- 1,706
26
votes
2
answers
3k
views
Corollaries of the Yoneda Lemma in Analysis?
This is a cross-post of my ~2 weeks (canonically) unanswered question on Math.SE: https://math.stackexchange.com/questions/1830287/corollaries-of-the-yoneda-lemma-in-analysis.
I am looking for some ...
- 2,680
25
votes
1
answer
3k
views
Does there exist a measurable function which is not a.e. "strongly" measurable?
More specifically, letting $I=[0,1]$, do there exist $f,E$ with $E$ a (necessarily nonseparable) Banach space and $f$ a bounded Lebesgue measurable function $I\to E$ such that $f$ is not equal almost ...
- 3,584
25
votes
2
answers
1k
views
Can nuclearity be determined by tensoring with a single C*-algebra?
A C*-algebra is nuclear if the algebraic tensor product $A\odot B$ ($B$ is any other C*-algebra) admits a unique C*-norm. This definition requires testing the condition for nuclearity with `all' C*-...
- 505
25
votes
2
answers
4k
views
Dual of the space of Hölder continuous functions?
Let $X=C^{\alpha}(\Omega,\mathbb{R})$ be the space of Hölder continuous functions. What is its dual?
- 1,256
25
votes
16
answers
4k
views
functions satisfying "one-one iff onto"
Hello Everybody.
I need some more examples for the following really interesting phenomenon:
A function from the class ... is one-one iff it is onto.
Some ...
24
votes
3
answers
3k
views
Can Hölder's Inequality be strengthened for smooth functions?
Is there an $\epsilon>0$ so that for every nonnegative integrable function $f$ on the reals,
$$\frac{\| f \ast f \|_\infty \| f \ast f \|_1}{\|f \ast f \|_2^2} > 1+\epsilon?$$
Of course, we ...
- 9,756
23
votes
2
answers
7k
views
What is a Gaussian measure?
Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.
Is there a direct ...
- 8,512
22
votes
2
answers
652
views
Does every positive continuous function have a non-negative interpolating polynomial of every degree?
Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
- 1,209
22
votes
5
answers
1k
views
Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$
Let $g\in C(\Bbb R)$ be given, we want to find a solution $f\in C(\Bbb R)$ of the equation
$$
f(x+1) + f(x) = g(x).
$$
We may rewrite the equation using the right-shift operator $(Tf)(x) = f(x+1)$...
- 1,245