Jochen Wengenroth
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Registered User
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I am Professor for Mathematics at the Unversität Trier (Germany)
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Jun 14 |
answered | Complete uniform spaces require complete metrics? |
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Jun 13 |
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Homeomorphisms and disjoint unions The authors also mention that one can modify their construction to obtain an example in $\mathbb R^2$. |
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May 17 |
answered | $C_c^{\infty}([0,T];V)$ is dense in $C_c^{1}([0,T];V)$? |
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May 17 |
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$C_c^{\infty}([0,T];V)$ is dense in $C_c^{1}([0,T];V)$? @MTS: $C_c^\infty(K) = \lbrace f\in C^\infty(\mathbb R): \mathrm{supp}(f) \subseteq K\rbrace$. |
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May 13 |
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Existence of dominating measure for weak*-compact set of measures Andy does not have a definition at all because his functionals $L_Z$ make sense only for bounded $Z$. Anyway, weak*-compactness could be obtained from the Banach-Alaoglu theorem. |
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Mar 21 |
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Smooth function algebra on cartesian product and beyond Just a little remark: That $\otimes_i$ coincides with $\otimes_pi$ in this case is not only because of nuclearity (your remark about the spaces of smooth functions with compact support shows this since $\mathscr D(\mathbb R) \tilde{\otimes}_\pi \mathscr D(\mathbb R) \neq \mathscr D(\mathbb R^2)$).You use first that $\otimes_i = \otimes_\varepsilon$ for Frechet spaces and then nuclearity. |
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Mar 19 |
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Riesz representation theorem for vector-valued fields @jbc: What do you mean by every Banach space is an inductive limit of its finite dimensional subspaces? In which category? Shouldn't the inductive limit $X=\lim X\alpha$ have the universal property that a linear map $X\to Y$ is continuous (a morphism of the category) iff all restrictions to $X_\alpha$ are continuous? If all $X_\alpha$ are finite dimensional this is no condition. |
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Mar 19 |
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Weak convergence in measure for negligible sets. For more information about the relation between weak and a kind of "set-wise" convergence look at the so-called Portmanteau theorem. |
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Mar 18 |
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Uniformly integrable sequence such that a.s. limit and conditional expectation do not commute @Didier: Thanks for editing. |
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Mar 16 |
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Extending a Hilbert space isometrically @Tom: Even if there is any metric on $F$ inducing the vector space topology such that $f$ is an isometry then $f(H)$ will be already a completely metrizable topological vector space. This not obvious (since the uniformities may be a priori different) but true because of a theorem of Victor Klee (solving a problem of Banach). Therefore, $F=f(H)$ still holds. |
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Mar 15 |
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Extending a Hilbert space isometrically ...normed and $f$ is an isometry then $f(H)$ is complete and hence closed in $X$ (if you assume that $X$ is Hausdorff). |
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Mar 15 |
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Extending a Hilbert space isometrically The obvious idea to embed the separable Hilbert space into a sequentially complete locally convex space is to find a sequence $x_n$ which converges fast to $0$ and to define $f:\ell_2 \to X$ by $f((a_n)_{n\in\mathbb N})= \sum\limits_{n=1}^\infty a_n x_n$. This map will be injective if the sequence is *topologically linearly m-independent$. This is discussed on page 37 in the book Barrelled locally convex spaces of Bonet and Perez-Carreras. The second question is not clear to me: If there is no norm on the Frechet space $F$, what do you mean by isometric? On the other hand, if $F$ is ... |
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Mar 15 |
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Uniformly integrable sequence such that a.s. limit and conditional expectation do not commute added 201 characters in body |
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Mar 15 |
answered | Uniformly integrable sequence such that a.s. limit and conditional expectation do not commute |
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Mar 14 |
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stopping time on event A $\lbrace f\in B\rbrace$ is a shorthand for $\lbrace \omega\in\Omega: f(\omega)\in B\rbrace$. The conventions for $\cap$ and $\cup$ are as for$\cdot$ and $+$. |
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Mar 14 |
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stopping time on event A Although this is not research level, $\lbrace \sigma' \le t \rbrace = \lbrace \sigma \le t \rbrace \cap A \cup \lbrace \tau \le t \rbrace \cap A^c \in F_t$ since $A\in F_\sigma \subseteq F_\tau$. |
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Mar 13 |
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A possible refinement of a theorem of Malliavin At first sight it seems to me that in Sotirov's thesis only sequential continuity is considered. Since $\mathscr D(\mathbb R^n)$ is very far from being metrizable this can be much weaker than contiunity. |
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Mar 12 |
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Exponential sums and binary expansions Very good. Thank you very much. There is a bounty on this question on math.stackexchange.com/questions/324496/… |
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Mar 12 |
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Exponential sums and binary expansions Your way to prove the identity is certainly better. |
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Mar 11 |
accepted | Inductive tensor product and smooth functions |
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Mar 11 |
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Inductive tensor product and smooth functions @Allan: You are right, my answer was not correct. I hope that the new answer is better. |
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Mar 11 |
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Inductive tensor product and smooth functions added 35 characters in body |
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Mar 11 |
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Exponential sums and binary expansions correced $n/2-1$ to $n/2 +1$ |
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Mar 11 |
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Inductive tensor product and smooth functions added 1885 characters in body |
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Mar 11 |
asked | Exponential sums and binary expansions |
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Mar 7 |
answered | Inductive tensor product and smooth functions |
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Mar 1 |
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a problem in functional analysis that erdos solved in 2 lines The transcendence of $\pi$ is due to to Bryant? |
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Feb 26 |
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A question of Allan on infinite divisibility Looks very good ($x$ should also belong to the generators). Thank you very much. |
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Feb 26 |
asked | A question of Allan on infinite divisibility |
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Feb 24 |
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Class of functions that the Fourier inversion holds The Fourier transform is an isomorphism also on $\mathscr S'(\mathbb R^d)$, the space of tempered distributions. |
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Feb 24 |
answered | spectacular applications of functional analysis in resolutions of apparently unrelated problems |
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Feb 21 |
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Compactly generated Banach spaces Yemon's answer shows that every separable Frechet space is compactly generated. The converse is also true: The compact generator is contained in the closed absolutely convex hull of a sequence converging to $0$, and the countable set of all rational (finite) linear combinations of that sequence is dense. |
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Feb 19 |
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Expectation of sample variance This is not research level. Nevertheless, under your assumptions, the empirical mean and variance are independent so that $E(S^2|\bar{X}=\bar{x})=E(s^2)$. |
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Feb 15 |
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When is a sequentially closed cone, closed? As for almost all locally convex properties, bornologicity does not reflect properties of single seminorms. The essential point is always the relation between them or how many of them you need. A trivial example: If only countably many seminorms describe the locally convex topology the space is (semi-) metrizable and hence bornological. |
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Feb 14 |
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When is a sequentially closed cone, closed? I do not understand this question. Barreledness is rather close to bornologicity, for instance, every (locally) complete bornological space is ultrabornological and hence barrelled. This means that barrelledness will not help you very much to conclude closed from sequentially closed. |
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Feb 14 |
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When is a sequentially closed cone, closed? added 158 characters in body |
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Feb 14 |
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When is a sequentially closed cone, closed? added 468 characters in body |
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Feb 13 |
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When is a sequentially closed cone, closed? Lieber Herr Michor, do you agree with the counterexample to your claim that I posted as an answer? |
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Feb 12 |
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When is a sequentially closed cone, closed? added 1 characters in body |
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Feb 12 |
answered | When is a sequentially closed cone, closed? |
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Feb 5 |
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An interesting summation It is very easy to calculate the derivative of $f(x)= \sum\limits_{k=1}^n \frac{x^k}{k}$. |
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Feb 5 |
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Can distribution theory be developed Riemann-free? Of course, if you want a characterization involving all Riemann integrable functions you have to use the Riemann integral. But continuous functions (or, by Weyl's criterion, exponentials) would be enough. |
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Feb 1 |
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Schwartz kernel theorem for topological spaces The answers of jbc and Peter Michor are more or less the same as both describe the $\varepsilon$-product $C(K_1) \varepsilon C(K_2)$ of Laurent Schwartz. |
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Feb 1 |
awarded | ● Yearling |
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Jan 30 |
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Second difference If you only want to have a fixed $x$ (as it seems since you accepted Xandi's answer) you can take any odd function $f$ with $f(0)=0$. |
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Jan 30 |
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Second difference Are you sure that there is a constant $C$ that works in this example for all other $x$? |
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Jan 30 |
answered | Metrization of weak convergence of signed measures |
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Jan 30 |
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An elementary probability question The question is probably about good bounds. |
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Jan 25 |
answered | Is an additive category a balanced category? |
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Jan 9 |
awarded | ● Organizer |
