jbc
|
Unregistered User
|
|
|
May 20 |
comment |
Existence of dominating measure for weak*-compact set of measures By "compatibility" above I of course meant "non-compatibility". It is perhaps worth mentioning that in the topological situation, the universal property works in the other direction. $S$ embeds into $M^t(S)$ in such a way that every continuous, bounded function on $S$ with values in a Banach space lifts in a unique fashion to a continuous linear mapping with the appropirate (which, again, is not the norm). |
|
May 20 |
comment |
Existence of dominating measure for weak*-compact set of measures Compatibility in the first part means that the corresonding dual spaces are too large. Thus the Banach space duals of the function spaces consist in both cases of the finitely additive measures, not the countably additive or Radon ones. The unversal property is that every countably additive meaure on the $\sigma$-algebra with values in a Banach space (for which see Diestel and Uhl "Vector measures") lifts to a unique continuous linear mapping on $L^\infty$ with the topology mentioned in my answer. |
|
May 20 |
revised |
Existence of dominating measure for weak*-compact set of measures corrected some irritating typos. |
|
May 20 |
answered | Existence of dominating measure for weak*-compact set of measures |
|
May 17 |
comment |
On uniform convergence of sequences of bounded holomorphic functions with formal convergence I can't give a reference but there is a large number of related results which follow from the following general considerations: the unit ball of $H^\infty$ is compact for the topology of compact convergence and so the latter coincides there with any weaker Hausdorff topology. In your case, this would be the weak topology induced by evaluation of the derivatives at $z_0$. |
|
May 17 |
comment |
Does the fact that this vector space is not isomorphic to its double-dual require choice? The precise reference for Garnir's result is to be found in MR0477688 (and the word "proved" is missing in the previous comment). |
|
May 17 |
comment |
Does the fact that this vector space is not isomorphic to its double-dual require choice? For the record, in 1973 the Belgian mathematician Henri Garnir, combining results of Schwartz on a measurable graph theorem and Solovay as mentioned here, that it is consistent with ZF without AC that every linear map from an ultrabornological space (in particular, Banach, Fréchet or an inductive limit of a sequence of Banach spaces which covers the spaces mentioned here) into any locally convex space is continuous. This in turn implies that the algebraic dual of such a space coincides with the topological dual and that the same holds for biduals for most spaces of interest. |
|
May 12 |
revised |
Applications of visual calculus 2 wee typos, one a wrong date |
|
May 12 |
answered | Applications of visual calculus |
|
May 12 |
comment |
Unit sphere in R^\infty is contractible? Sorry. This was meant to be a comment on the above comment. Tried to delete but couldn't. Maybe somebody more powerful could erase it. |
|
May 12 |
answered | Unit sphere in R^\infty is contractible? |
|
May 12 |
answered | Is there a good reason why a^{2b} + b^{2a} <= 1 when a+b=1? |
|
May 8 |
accepted | radon-nikodým property of $\ell^\infty$ |
|
May 7 |
answered | radon-nikodým property of $\ell^\infty$ |
|
Apr 28 |
accepted | Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spaces |
|
Apr 27 |
answered | Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spaces |
|
Apr 23 |
accepted | Extending uniformly continuous functions on subspaces to non-metrizable compactifications |
|
Apr 23 |
comment |
Gelfand representation and functional calculus applications beyond Functional Analysis My understanding of the proof is that Carleson demonstrated the validity of the interpolation property with hard analysis (which, by the way, led to the important concept of Carleson measures). The Banach algebra bit, i.e., the proof of the equivalence of this with the denseness (which is apparently due to Newman), uses the fact that the Gelfand-Neumark transform allows one to consider bounded analytic functions on the open disc and their absolute values as continuous functions on a compactum, the point being that such functions attain their infimum. |
|
Apr 23 |
comment |
Extending uniformly continuous functions on subspaces to non-metrizable compactifications If you have $Y$ and $Z$ as in your question with the extension property for each subspace $X$ and each bounded, uniformly continuous function thereon, then by using $X=Y$ one sees that $Z$ has the uiversal property for each bounded, uniformly continuous function on $Y$. This means that $Z$ is the so-called Samuel compactification of $Y$. So the only leeway is in the case of $Y$ for which the two notions of compactification diverge. |
|
Apr 23 |
answered | Extending uniformly continuous functions on subspaces to non-metrizable compactifications |
|
Apr 22 |
answered | Gelfand representation and functional calculus applications beyond Functional Analysis |
|
Apr 20 |
comment |
Meromorphic Functions as Distributions Yes, indeed it does. |
|
Apr 18 |
comment |
Meromorphic Functions as Distributions I forgot one more ingredient---you can always multiply a distribution with a smooth function. So you consider an open cover which is such that each element contains only one pole. The restriction of the meromorphic function to such a set is an analytic function times one of the form $(x-x_0)^{-n}$ and so a distribution. Now globalise. |
|
Apr 18 |
answered | Meromorphic Functions as Distributions |
|
Apr 15 |
revised |
Does the derivative of log have a Dirac delta term? small addition on why the logarithm of the absolute value is a distribution |
|
Apr 15 |
answered | Does the derivative of log have a Dirac delta term? |
|
Apr 12 |
answered | Fixed point theorems |
|
Apr 11 |
answered | A graduate course on Sturm Liouville theory? |
|
Apr 11 |
comment |
Fixed point theorems Another contribution to the theme "FTP's and Nobel Prizes in economics". The Arrow-Debreu theory of equilibrium in economics uses the Brouwer FTP and its extension by Kakutani in an essential way. Both are laureates and this theory is generally regarded as one of their most significant contributions. |
|
Apr 11 |
comment |
Fixed point theorems It is worth mentioning the sensationally short proof given by Lomonosov of his theorem that every continuous linear mapping on a Banach space which commutes with a non-zero compact operator has a non-trivial invariant subspace. This was then the strongest positive result on the invariant subspace problem (and might still be for all I know) and the key ingredient was the Schauder-Tychonoff FTP. |
|
Apr 11 |
comment |
Fixed point theorems Another application is a nice proof of the Inverse Function Theorem. |
|
Apr 10 |
revised |
Fixed point theorems edited to comply with the terms of reference of the OP. |
|
Apr 10 |
answered | Fixed point theorems |
|
Apr 10 |
comment |
Fixed point theorems Not a FPT but a book: "Fixed point theory" by Granas and Dugundji. |
|
Apr 7 |
revised |
Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions? two minor typos |
|
Apr 4 |
comment |
Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$? The answers below show that this is false but it is perhaps worth mentioning that, on the positive side, there are versions under additional conditions which are correct and these are frequently used in the isomorphic theory of Banach spaces. They go under the collective name of "Pelczynski decomposition method". |
|
Apr 4 |
answered | Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions? |
|
Mar 24 |
accepted | Riesz representation theorem for vector-valued fields |
|
Mar 24 |
comment |
The right conformal map to make a certain picture Not directly relevant but you might be interested in the article "Conformal and equivalent world maps" by B. H. Brown in the Amer. Math. Monthly. vol. 42. As the title suggests, it is motivated by (two central) problems of mathematical cartography but the main results are purely mathematical. The relevant one is an explicit description of all conformal mappings of the plane which map the rectangular array onto systems of conic sections. (He also solves the same problem for area-preserving mappings). |
|
Mar 23 |
revised |
Measurable functions and unbounded operators in von Neumann algebras corrected two small typos |
|
Mar 23 |
answered | Measurable functions and unbounded operators in von Neumann algebras |
|
Mar 23 |
answered | Riesz representation theorem for vector-valued fields |
|
Mar 22 |
answered | Convergence of probability measure and the *-weak convergence ? |
|
Mar 22 |
comment |
Riesz representation theorem for vector-valued fields In the category of Banach spaces with linear contractions as morphisms. It is just a fancy way of saying that any Banach space is a union of its finite dimensional subspaces and that a compatible family of contractions on these subspaces combine to a contraction on the whole space. I didn't really expect to have to justify such a simple fact in detail. I was just making a side remark in a comment, not writing a mathematical journal article. |
|
Mar 21 |
comment |
Functional Analysis Generalizations: indeterminated inner product and functions over manifolds Infinite dimesional spaces with indefinite metrics have been studied by the Ukrainian school of operator theory (Krein). See, for example, "Hilbert space with an indefinite metric" by Nikolskii. |
|
Mar 18 |
comment |
Riesz representation theorem for vector-valued fields The best (relatively recent) reference for this circle of ideas is the classic "Vector measures" by Diestel and Uhl and I would recommend this as a starting point. The standard treatise of Dunford and Schwartz also contains relevant material. If $V$ is a Banach space and $Q$ is compact, one can indeed get a result of this form using methods of category theory by using the fact that every Banach space is an inductive limit of its finite dimensional subspaces and then dualising---for a suitable definition of measures with values in a dual Banach space. This can then be generalised. |
|
Mar 15 |
comment |
Extending a Hilbert space isometrically Since you can always embed $H$ isometrically into $H \times Y$ for any topological vector space $Y$, there would seem to be plenty of scope. |
|
Mar 11 |
comment |
Is $C^{\infty}[0,1]$ or $S$ separable? Since we are looking at this situation in more detail, we might as well go the whole hog. In a short and elegant paper (Math. Ann. vol. 164), Pietsch showed that for a self-adjoint operator on Hilbert space (not necesarily bounded), the intersection of the domains of definition of its powers is a Fréchet space in a natural way. He gives three main examples and these are precisely the three which appear here. The resulting space is separable (if the Hilbert space is) and he gives simple characterisations of when it is Montel or nuclear, using growth conditions om the eigenvalues. |
|
Mar 10 |
comment |
Is $C^{\infty}[0,1]$ or $S$ separable? If one wants a more direct proof which uses the Weierstrass theorem, one can proceed as follows: there is a standard method to embed $C^\infty(I)$ as a closed subspace of a countable product of copies of $C(I)$ (simply map $f$ onto the sequence $(f^{(n)}$ of its derivatives). The latter is separable. |
|
Mar 10 |
comment |
Is $C^{\infty}[0,1]$ or $S$ separable? Both are nuclear Fréchet spaces and both have Schauder bases, each of which imply separability (reference: "Nuclear locally convex spaces" by Albrecht Pietsch, for example). |
