Martin
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Registered User
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3h |
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Importance of separability vs. second-countability What makes the simple fact that images of separable spaces are separable an important theorem? |
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May 2 |
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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. |
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Apr 27 |
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Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spaces Does 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 $\beta (D_1 \times D_2) \neq \beta D_1 \times \beta D_2$. |
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Apr 25 |
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separable spaces-QM vs. functional analysis Since $\psi_n$ 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 $\psi_n$. |
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Apr 24 |
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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. |
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Mar 29 |
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if $S \times \Re$ is diffeomorphic to $T \times \Re$ then are S and T diffeomorphic? No worries, I removed my comment :-) |
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Mar 26 |
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Are there results in “Digit Theory”? There is some debate on the last paragraph on meta: meta.mathoverflow.net/discussion/1561 |
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Mar 26 |
awarded | ● Critic |
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Mar 24 |
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What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$? The Banach algebra $L^1(G)$ determines $G$ and its topology by Wendel's theorem: math.stackexchange.com/q/328443. This is not true for $C_{r}^\ast(G)$, so it seems that the question in the title is the wrong way around and the last sentence is the right way of looking at it. |
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Mar 20 |
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Product of Baire sigma-algebras NB: 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 $M \notin \mathcal{E} \otimes \mathcal{E}$. |
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Mar 20 |
answered | Product of Baire sigma-algebras |
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Mar 17 |
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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". |
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Feb 27 |
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The point of view of semicats in functional analysis Thanks 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. |
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Feb 27 |
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The point of view of semicats in functional analysis No 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. |
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Feb 26 |
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The point of view of semicats in functional analysis What 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? |
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Feb 23 |
awarded | ● Enlightened |
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Feb 23 |
accepted | Is the closed unit ball of the Hilbert space homeomorphic to the unit sphere ? |
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Feb 20 |
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How to see such space is Lindelof? On math.SE you were linked to Dan Ma's topology blog dantopology.wordpress.com/2012/10/30/… where there is a proof in the first paragraph on "Non-normal product spaces". |
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Feb 13 |
awarded | ● Nice Answer |
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Feb 13 |
answered | Is the closed unit ball of the Hilbert space homeomorphic to the unit sphere ? |
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Feb 11 |
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Complexifying a real Banach space and its dual I 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. |
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Feb 10 |
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pointwise ergodic theorem and mean sojourn time Link to the question on math.SE: math.stackexchange.com/q/297233 |
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Feb 10 |
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Must the Minkowski sum of a Borel set and a *closed* ball be Borel? On the duplicate thread on math.SE math.stackexchange.com/q/298494 user 5PM asked: "The answer by Tapio Rajala applies to $n \geq 3$ only. What if $n=2$? (The case n=1 has an easy affirmative answer)." |
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Feb 9 |
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isometric embeddings of Cayley graphs in “nice” spaces Thank you. I found the article: W. Holsztyński, $\mathbf R^n$ as a universal metric space, Notices AMS 25 (3) (1978) A- 367. |
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Feb 9 |
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isometric embeddings of Cayley graphs in “nice” spaces Could 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. |
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Feb 2 |
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Fredholm and Compact Operators Also asked on math.SE: math.stackexchange.com/q/293019 where this question seems to be a better fit. |
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Feb 2 |
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Equivariant forms and localization Same question on SE: math.stackexchange.com/q/291408/49437 |
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Feb 2 |
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Equivariant integration (localization formula) Same question on SE: math.stackexchange.com/q/291566/49437 |
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Jan 30 |
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Metrization of weak convergence of signed measures It 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 $\{0\}$. |
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Jan 28 |
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Behaviour of power series on their circle of convergence The 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. |
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Jan 27 |
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Showing a Banach space is reflexive The 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. |
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Jan 15 |
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Algebraic Morse theory Also on math.stackexchange math.stackexchange.com/q/278461 |
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Jan 6 |
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definition of operator valued integral with spectral measure Also on stackexchange: math.stackexchange.com/q/270581 |
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Jan 6 |
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Absolute norms and 1-unconditional sums Also on stackexchange: math.stackexchange.com/q/271686 |
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Jan 3 |
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Baire sets of $X$ possess the required Cartesian product property math.stackexchange.com/q/268533 |
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Dec 31 |
awarded | ● Nice Answer |
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Dec 30 |
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Old books still used I'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 |
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Dec 29 |
revised |
Old books still used deleted 10 characters in body |
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Dec 29 |
answered | Old books still used |
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Dec 6 |
awarded | ● Commentator |
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Dec 5 |
awarded | ● Supporter |
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Dec 3 |
awarded | ● Citizen Patrol |
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Dec 3 |
answered | Degeneracies for semi-simplicial Kan complexes |
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Nov 30 |
awarded | ● Teacher |
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Nov 30 |
awarded | ● Editor |
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Nov 30 |
revised |
Product of Borel sigma algebras added 33 characters in body |
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Nov 30 |
answered | Product of Borel sigma algebras |
