All Questions
966 questions
6
votes
1
answer
754
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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
2
answers
1k
views
The contractivity of the heat semigroup in $L^p$ spaces
Let $M$ be a Riemannian manifold. By functional calculus, it is immediate to show that the heat semigroup is a contraction in $L^2(M)$. I can also show that it is a contraction in any $L^p(M)$ with $p ...
- 5,407
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
$$
\int_{x\in X}\,d(y_0,f(x)...
- 5,405
6
votes
0
answers
241
views
Extension of positive functionals
Let $X$ be a function space as $C(K)$ or $L^p$, with its usual norm and order, that is $f \le g$ if and only if $f(x) \le g(x)$ for a.e. $x$. If $M$ is a subspace of $X$ and $L:M \to \bf R$ is a ...
- 6,035
6
votes
3
answers
852
views
Are nuclear operators closed under extensions?
Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram
$\require{AMScd}$
\begin{CD}
0 @>>> X_1 @>f_1>> X_2 ...
- 1,338
6
votes
1
answer
498
views
A finely open set, not open up to polar set?
I already asked this on M.SE, but get no answers.
Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge 2$, which is not ...
- 1,724
6
votes
1
answer
538
views
Growth rate of the infinity norm of Discrete Fourier Transform of +1,-1 vectors
Let $f=(f_0,\ldots,f_{n-1})$ be a vector in $V_n=\{\pm 1\}^n$. Let
$F=(F_0,\ldots,F_{n-1})$
be its (discrete) Fourier transform defined by
$$
F_k=\sum_{x=0}^{n-1} f_x \omega_n^{x k}
$$
where $\...
- 10.4k
6
votes
2
answers
324
views
Nonvanishing section of infinite-dimensional tautological bundle
Let $H$ be a real or complex Hilbert space. In the case where $H$ is infinite-dimensional, let us define a half-dimensional subspace as a subspace $W \subset H$ such that both $W$ and $W^\perp$ have ...
- 9,853
6
votes
1
answer
641
views
Bochner-Minlos for moment-generating functions?
It is well-known that the Bochner-Minlos theorem characterises measures on duals of nuclear spaces by their characteristic functions. Is there a similar version for moment-generating functions?
I have ...
- 651
6
votes
1
answer
574
views
English translation of Schwartz's papers on vector-valued distributions
I am interested in systematically studying the theory of vector-valued distributions. The original two papers due to Laurent Schwartz entitled Théorie des distributions à valeurs vectorielles. I & ...
- 183
6
votes
1
answer
455
views
Is the tensor product of distributions a continuous bilinear map with respect to the weak topology?
Let $X$ and $Y$ be smooth manifolds. The map $\mathcal{D}'(X)\times\mathcal{D}'(Y)\to\mathcal{D}'(X\times Y)$ given by $(S,T)\mapsto S\boxtimes T$ is continuous with respect to the strong topology. Is ...
- 387
6
votes
2
answers
460
views
Terminology: Banach spaces equipped with continuous associative product?
This is admittedly a low-interest question mathematically, and is arguably a question I could resolve if I had time over the next few days to go and look through a large number of the Banach algebra/...
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
1
answer
549
views
Volume doubling, uniform Poincaré, counterexample
The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates.
Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...
- 721
6
votes
1
answer
294
views
Idempotent functions on Sp(1)
The quaternion group $Sp(1)\simeq S^3$ can be understood as $(z,w)\in\mathbb {C}^2$ with $|z|^2+|w|^2=1$ where multiplication is defined by $(z,w)(t,s)=(zt-\bar{s}w,zs+\bar{t}w)$.
Question: How do ...
- 1,583
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
236
views
Potential p-norm on tuples of operators
Consider $\left[\begin{matrix}A \\ B\end{matrix}\right] \in B(H)^2$. One can define
$$
\left\|\left[\begin{matrix}A \\ B\end{matrix}\right]\right\|_p = \| |A|^p + |B|^p\|^{1/p}.
$$
Q: Is this a norm?
...
- 3,984
6
votes
0
answers
529
views
Infinite-dimensional "algebraic varieties"
This question was also formerly posted on MSE but has not received any answer or comment.
Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its projectivization. ...
- 1,543
6
votes
1
answer
693
views
Do multiplicative Banach limits exist?
Let $(D, \succeq)$ be a directed set, and let $B$ be the space of real-valued bounded functions on $D$. A Banach limit $\ell$ on $D$ is a linear functional that satisfies
$$\sup_{d \in D} \inf_{c \...
- 869
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:
$$
\mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...
- 1,090
6
votes
1
answer
798
views
Prohorov's theorem for random elements of Hilbert space: weak convergence
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(E,\mathcal{E})$ be a separable Hilbert space ($E$) with Borel $\sigma$-algebra $\mathcal{E}$. For concreteness let us set $E=L^{2}[a,b]$ ...
- 61
6
votes
2
answers
539
views
Is there a reasonable notion of spectral theorem on a pre-Hilbert space?
I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...
- 269
6
votes
1
answer
422
views
Absolutely 2-summable operator on a Hilbert space
An bouneded linear operator $A \in L(X, Y)$ (here $X$, $Y$ are Banach spaces) is called absolutely $2$-summable if there exists a $C>0$ such that
$$ \left( \sum_{j=1}^N \| A x_j\|_X^2 \right)^{1/2} ...
- 9,853
6
votes
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
2
answers
529
views
Schrödinger eigenfunctions are bounded
Let $V:\mathbb{R}\rightarrow \mathbb{R}^{+ *}$ a real positive function such that $\displaystyle \lim_{ x \to \pm\infty} V(x)= +\infty $.
Then the Schrödinger operator $H=-\frac{d^2}{dx^2}+V(x)$ has ...
- 129
6
votes
1
answer
822
views
Variations on the Mellin and Dirichlet transforms
There are a number of variations on the Laplace transform that turn up all over math. Some examples:
$\int_{-\infty}^{\infty} f(t)e^{-st} dt$ - The Laplace transform
$\sum_{-\infty}^{\infty} f(t)z^{-...
- 4,897
6
votes
1
answer
583
views
Set of w*-continuous operators closed for the weak* topology or not?
Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...
user64615
6
votes
2
answers
571
views
Why are $\Gamma_0$ functions called this
It is very common to indicate with $\Gamma_0(A)$ the set of lower semicontinuous convex functions from $A$ to $(-\infty,+\infty]$ with nonempty domain. An example of usage of this notation can be ...
- 71
6
votes
1
answer
773
views
When do Borel $\sigma$-algebras generated by the total variation norm and the weak* topology coincide?
I am almost certain that I read somewhere that the following is true, but I cannot seem to locate the reference. I would be most appreciative if someone could point me to a reference. The result was ...
user12713
6
votes
0
answers
798
views
What is the Banach dual of the Bochner space $L^\infty(\Omega;X)$?
Suppose $\Omega$ is a $\sigma$-finite measure space (I'm happy to take $\Omega = \mathbb{N}$) and let $X$ be a Banach space. It's pretty well known that the Banach dual of $L^\infty(\Omega)$ can be ...
user14166
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
965
views
C*-algebras and quantum fields
One can represent a quantum system by the Weyl algebra (which is a C*-algebra). For instance, a 1 degree of freedom system can be represented by the algebra generated by $e^{\imath t Q}, e^{\imath s P}...
- 585
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
5
votes
0
answers
222
views
Right derived contravariant hom-functor
I am interested in additive categories appearing in functional analysis, in particular, the category $LCS$ of locally convex spaces and continuous linear functions. This category is not abelian but ...
- 16.4k
5
votes
1
answer
564
views
Convergence of discrete Laplacian to continuous one
I make the following observation:
Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.)
This one has eigenvalues ...
- 536
5
votes
0
answers
348
views
Discrete groups G whose full C*-algebra C*(G) is not MF?
This is a cheap rip-off of this question, but I am genuinely interested in an answer.
Is there a known example of a countable discrete group G whose full group C*-algebra C*(G) is not MF?
Let us ...
- 1,786
5
votes
1
answer
2k
views
Commuting with self-adjoint operator
Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$
My thought was that using a ...
- 501
5
votes
0
answers
315
views
Schauder basis in the Arens-Eells space
Context
Arens-Eells space. Let $X$ be a separable pointed metric space with base point $e$. An elementary molecule is defined as follows (Nik Weaver, Lipschitz Algebras, 2nd ed.)
$$
m_{pq} := \delta_p ...
- 437
5
votes
2
answers
516
views
Biorthogonal functionals
If $X$ is a separable Banach space and $(x_n)$ is a basic sequence, then we can define biorthogonal functionals $(x^{*}_n)$ in $X^{*}$ such that $x^{*}_n(x_k)=\delta_{nk}$.
What about conversely? If ...
- 1,361
5
votes
1
answer
2k
views
Dual of the space of continuous functions
Let $T \subseteq \mathbb R$ be a closed set of real numbers. Let $X := C(T, \mathbb R)$ denote the Fréchet space of continuous real-valued functions on $T$. The topology on $X$ is generated by ...
- 8,512
5
votes
1
answer
878
views
About a reparable error discovered in a submitted article
What can I do if I discover a reparable error in an article that I submitted to a journal of mathematics? I want to know the possible solutions so that the article is not rejected. And what happens if ...
5
votes
1
answer
284
views
Malliavin derivative of stopped Brownian motion
Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"
I have a small question concerning the Malliavin derivatives. It could ...
- 393
5
votes
2
answers
358
views
Linear transport equation with unbounded coefficients
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its ...
- 124
5
votes
1
answer
602
views
Invariant probability on a unit ball of a Banach space
Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries.
Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) ...
- 53
5
votes
0
answers
350
views
How to calculate the volume of a parallelepiped in a normed space?
Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
- 5,529
5
votes
2
answers
606
views
Banach-Mazur distance to complex $\ell^1$ of a space containing real $\ell^1$
Consider a complex Banach space $X$ with a real subspace isometric to $\ell^1_{\mathbb R}$. What is the best constant $c$ such that $X$ contains a complex subspace $c$-isometric to $\ell^1_{\mathbb C}$...
- 9,486
5
votes
1
answer
148
views
A perturbation of an unconditionally convergent series in $\ell_2$
For two functions $x,y:\omega\to\mathbb R$ let $xy:\omega\to\mathbb R$, $xy:n\mapsto x(n)y(n)$, be their pointwise product.
It is clear that for any elements $x,y\in\ell_2$ their pointwise product $xy$...
- 41.9k
5
votes
1
answer
2k
views
Are piecewise linear functions dense in $W^{1,\infty}$?
Are piecewise linear functions dense in $W^{1,\infty}$ ?
- 393
5
votes
2
answers
418
views
A question about Schwartz-type functions used in analytic number theory
In analytic number theory we like to weigh our counting functions with a smooth function $f$, so that we may apply Poisson's summation formula and take advantage of Fourier transforms. Typically the ...
- 26.9k
5
votes
1
answer
391
views
Is this closed subspace of Fréchet space complemented
In the hope of completing the rich tapestry of complemented (or not) topological vector subspaces, I would like to know (maybe it is immediate for specialists)
whether the space of analytic functions ...
- 3,738