# Martin

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Name        Martin
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3h     comment   Importance of separability vs. second-countabilityWhat makes the simple fact that images of separable spaces are separable an important theorem?
May2   comment   Reason for studying coherent sheaves on complex manifolds.I think it is due to using * which confuses the renderer because it wants to set things between * as italics. If  you replace * by \ast everywhere, then the preview looks fine.
Apr27  comment   Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spacesDoes the following work? If $X_1$ and $X_2$ are not totally bounded, they contain $\varepsilon$-discrete infinite subsets $D_1$ and $D_2$ for some $\varepsilon$. Their closures in the Samuel compactifications should be $\beta D_1$ and $\beta D_2$. Moreover, $D_1 \times D_2$ is $\varepsilon$-discrete, so its closure in the Samuel compactification of $X_1 \times X_2$ is $\beta (D_1 \times D_2)$. But $\beta (D_1 \times D_2) \neq \beta D_1 \times \beta D_2$.
Apr25  comment   separable spaces-QM vs. functional analysisSince $\psi_n$ is supposed to be a Cauchy sequence, it is contained in some ball of radius $R$ around zero. Assuming $R \gt \varepsilon$, no vector $\psi$ of norm $\geq 2R$ can be near any $\psi_n$.
Apr24  comment   separable spaces-QM vs. functional analysis"There exists a Cauchy sequence ..." seems to be a typo (since Cauchy sequences are bounded, the condition is obviously nonsensical). Delete "Cauchy" and you get a dense sequence $\psi_n$, hence a countable dense set. For the other direction enumerate the countable dense set to get a sequence satisfying Zettili's condition.
Mar29  comment   if $S \times \Re$ is diffeomorphic to $T \times \Re$ then are S and T diffeomorphic?No worries, I removed my comment :-)
Mar26  comment   Are there results in “Digit Theory”?
Mar26  awarded   ● Critic
Mar24  comment   What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?
Mar20  comment   Product of Baire sigma-algebrasNB: this also tells us how to find a meager set that doesn't belong to $\mathcal{E} \otimes \mathcal{E}$: take a universal analytic set $A$ in $X \times X$ and take an open set $U$ such that $M = A \mathbin{\Delta} U$ is meager. Then $M \notin \mathcal{E} \otimes \mathcal{E}$.
Mar20  answered  Product of Baire sigma-algebras
Mar17  comment   Compactness of sigma-algebra for the $L^1$ metrics@Didier Piau: Dunford and Schwartz, Part I, Section III.7 has a paragraph The metric space $\Sigma(\mu)$ plus some exercises in III.9. It appears in the index "measure space, as a metric space".
Feb27  comment   The point of view of semicats in functional analysisThanks for the clarification. You could be interested in reading about operator ideals. These give a vast number of interesting functional-analytic examples of "semicats" different from the compact operators.
Feb27  comment   The point of view of semicats in functional analysisNo immediate point, really. I suppose I'm puzzled about the beating around the bush with monics and epics in title, body of the question and the comments, while the question, as you say, asks something different.
Feb26  comment   The point of view of semicats in functional analysisWhat is the difficulty in determining monics and epics? If an operator has non-trivial kernel, it kills some map from $\mathbb{R}$ and if the range is not dense, Hahn-Banach yields a non-zero functional vanishing on the range. Operators with finite-dimensional source or target are compact. This hardly needs a reference, does it?
Feb23  awarded   ● Enlightened
Feb23  accepted  Is the closed unit ball of the Hilbert space homeomorphic to the unit sphere ?
Feb20  comment   How to see such space is Lindelof?
Feb13  awarded   ● Nice Answer
Feb13  answered  Is the closed unit ball of the Hilbert space homeomorphic to the unit sphere ?
Feb11  comment   Complexifying a real Banach space and its dualI believe the standard way is due to Dieudonné: dx.doi.org/10.1090/S0002-9939-1952-0047252-8 where he also proves that the James space is not the underlying real space of a complex Banach space thus disproving a conjecture of Banach. I think Ivan Singer's Bases in Banach spaces, I contains a discussion of the complexification in quite some detail on the first few pages.
Feb10  comment   pointwise ergodic theorem and mean sojourn time
Feb10  comment   Must the Minkowski sum of a Borel set and a *closed* ball be Borel?
Feb9   comment   isometric embeddings of Cayley graphs in “nice” spacesThank you. I found the article: W. Holsztyński, $\mathbf R^n$ as a universal metric space, Notices AMS 25 (3) (1978) A- 367.
Feb9   comment   isometric embeddings of Cayley graphs in “nice” spacesCould you please give a more precise reference to your short note? It is impossible for me to guess either title of the paper or its author from the information on this page.
Feb2   comment   Fredholm and Compact Operators
Feb2   comment   Equivariant forms and localization
Feb2   comment   Equivariant integration (localization formula)
Jan30  comment   Metrization of weak convergence of signed measuresIt seems easier to argue that $X^\ast$ has uncountable dimension as a vector space. Every weak*-neighborhood contains a linear subspace of finite codimension, so the intersection of countably many $0$-neighborhoods contains a subspace of countable codimension, in particular it can't be reduced to $\{0\}$.
Jan28  comment   Behaviour of power series on their circle of convergenceThe question came up on SE again: math.stackexchange.com/q/288765 user mrf refers to Lukašenko S. Ju., Sets of divergence and nonsummability for trigonometric series, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1978, no. 2, 65–70 for the result that there is a $G_\delta$-set which is not a set of convergence.
Jan27  comment   Showing a Banach space is reflexiveThe examples you mention are very easy to recognize as non-reflexive. $X$ is reflexive iff $X^\ast$ is reflexive and closed subspaces of reflexive spaces are reflexive. Identify $(c_0)^\ast = \ell_1$ and $(c_0)^{\ast\ast} = \ell_\infty$. Thus, $c_0$ is not reflexive. It follows that $\ell_1$ and $\ell_\infty$ are not reflexive either. Now show that you can embed $c_0$ as a closed subspace into a space of (continuous) bounded functions or $\ell_1$ into the dual of such a space and this covers all of your examples.
Jan15  comment   Algebraic Morse theory
Jan6   comment   definition of operator valued integral with spectral measure
Jan6   comment   Absolute norms and 1-unconditional sums
Jan3   comment   Baire sets of $X$ possess the required Cartesian product property
Dec31  awarded   ● Nice Answer
Dec30  comment   Old books still usedI'm really surprised at the suggestion that a book first published in the 1980ies should be a serious contender for "oldest books regularly used" in classical analysis
Dec29  revised   Old books still useddeleted 10 characters in body
Dec29  answered  Old books still used
Dec6   awarded   ● Commentator
Dec5   awarded   ● Supporter
Dec3   awarded   ● Citizen Patrol
Dec3   answered  Degeneracies for semi-simplicial Kan complexes
Nov30  awarded   ● Teacher
Nov30  awarded   ● Editor
Nov30  revised   Product of Borel sigma algebrasadded 33 characters in body
Nov30  answered  Product of Borel sigma algebras
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