2,465 reputation
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bio website math.uni-trier.de/~wengenroth
location Universität Trier
age 47
visits member for 2 years, 10 months
seen yesterday
I am Professor for Mathematics at the Unversität Trier (Germany)

1d
comment density of penalizations of Gaussian probability measures
$\mathscr S'(\mathbb R)$ is a very good locally convex space (the dual of a nuclear Frechet space), in particular it is the union of an increasing family of compact sets. Doesn't this imply that every probability measure on it can be approximated by a compactly supported one? Perhaps Minlos' theorem has some relevance.
2d
comment Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?
Well, well,... I consider "the rest of the proof" as the main point of the OMT. In Rudin's book, this is even part of the statement.
Dec
8
comment Can one show that the dual of a quasi-Banach space separates points without explicitly identifying the dual?
A trivial sufficient condition is that $X$ embeds continuously into some Hausdorff locally convex space $Y$.
Dec
8
comment Is every sigma-algebra the Borel algebra of a topology?
@Freeze_S No, it is the first uncountable ordinal.
Dec
1
comment Defining density of a random function using Radon-Nikodym Theorem
To stimulate this thread you should try to say more precisely what you are looking for. As stated, your question does not make much sense (you could take $\mu=P_X$ and density $1$). For $E=C([0,\infty))$ the Wiener measure (distribution of Brownian motion) is certainly an interesting measure on $E$, but without more context it is not clear whether this is an interesting case for you.
Dec
1
comment Bounded operator on a normed space with empty spectrum
The general fact Alexander is referring to is sometimes called "abstract Mittag-Leffler theorem" (a version of this is e.g. in Bourbaki). The first explicit appearance however is in an article of R. Arens from 1958.
Dec
1
comment Extend product sigma-algebra to cross-constant sets
Isn't it just the axiom of choice which excludes a "Lebesgue measure " on the power set of $[0,1]$?
Nov
28
comment Defining density of a random function using Radon-Nikodym Theorem
Do you mean that $E$ is a Banach space of real-valued functions on $[0,1]$ (like $E=C([0,1])$) and $X:\Omega \to E$ is a process with paths in $E$? Anyway, if you do not have a reference measure on $E$ it makes no sense to speak about "the density".
Nov
25
comment decomposition of tempered distributions by entire analytic functions
I guess you mean by $g^\vee$ the inverse Fourier transform. Since Fourier transformation is an isomorphism $\mathscr S'\to\mathscr S'$ (with the weak*-topology) you need to show that $\sum\limits_{k=1}^K \phi_k \hat f$ converges to $\hat f$ and hence $\langle \sum\limits_{k=1}^K \phi_k \hat f,g \rangle \to \langle \hat f ,g\rangle$ for all $g\in\mathscr S$ (where$\langle f,g \rangle$ denotes the duality between $\mathscr S'$ and $\mathscr S$). For this it would be enough to show $\sum\limits_{k=1}^K \phi_kg \to g$ in $\mathscr S$ and this will depend on bounds for the derivatives of $\phi$.
Nov
25
comment Continuity of a Functional
The question is not clear. What is the domain of $T$? Do you mean by $F^{-1}(s)$ the multiplicative inverse $1/F(s)$?
Nov
24
awarded  Nice Answer
Nov
20
comment Between compact and locally uniform: What is the name of this convergence?
Very informative answer. A reference would be nice.
Nov
19
comment Universal maps between topological spaces
@Oblomov Surjectivity is clearly necessary but universality is much stronger (the identity on $\mathbb R$ is surjective but not universal with $f(x)=x+1$).
Nov
16
comment When is a `1-form' with continuous coefficients exact?
Very long second...
Nov
11
comment Counterexample for closed graph theorem in unmetrizable case
For an infinite-dimensional Hilbert space $X$ consider $Y=X$ endowed with the finest locally convex topology and the identical map. On the other hand, if the domain $X$ is ultrabornologial the closed graph theorem holds for so-called webbed spaces introduced by de Wilde. This is a very large class of locally convex spaces which contains all Banach spaces and is stable w.r.t. closed subspaces, separated quotients, and countable products and direct sums. See, e.g., Introduction to Functional Analysis of Meise and Vogt.
Nov
5
comment Rank of a sequence of covariance matrices
In general, $ZX_i$ are only in $L^1(\Omega,P)$ so that the covariance is not necessarily defined. $Z\in L^\infty$ would be okay.
Oct
31
comment “Partition” of a smooth function in $\mathbb R^2$
@GHfromMO Formally, you are right. The statement means: For all $f\in C^\infty(\mathbb R^2)$ with $f(0,0)=0$ there are $g_1,g_2 \in C^\infty(\mathbb R^2)$ such that $f(x,y)=g_1(xy,x)+g_2(xy,y)$ for all $(x,y)\in\mathbb R^2$.
Oct
30
comment Large Deviations: Exponential decay in normed spaces
There is a Banach-space version of Azuma's inequality due to Assaf Naor, theorem 1.5 in [web.math.princeton.edu/~naor/homepage%20files/AR-notes.ps]. There you need however special properties of the norm (modulus of smoothness of power type 2).
Oct
28
comment Examples of topologies compatible with a given dual pair
The answer to your question depends very much on what you mean by concretely. If the strong topology $\tau$ on $\mathscr E'$ (Schwartz' notation for $C^\infty(\mathbb R)'$) is concrete enough you can restrict it to $\mathscr D= C^\infty_c(\mathbb R)$ to obtain a compatible topology for the dual pair.
Oct
28
comment Examples of topologies compatible with a given dual pair
You mean $\langle\varphi,f\rangle = \int \varphi(x)f(x)dx$, right?