Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifolds Infinite dimensional differential geometry Manifolds of mappings Further applications References Index.

/pdf/the-convenient-setting-of-global-analysis-378u9vden9.pdf

The Convenient Setting of Global Analysis

We show that every Frechet differentiable real function onC(K), K scattered with locally uniformly continuous derivative has locally compact derivative. Using this and similar results, we investigate the existence ofC2-Frechet smooth surjections between various Banach spaces.

/pdf/smooth-functions-onc-k-3yuq49c3k2.pdf

Smooth functions onC(K)

https://dialnet.unirioja.es/descarga/articulo/118398.pdf

Compact polynomials between Banach spaces

Let $E$ be a Banach space and, for any positive integer $n$, let ${\cal P}(^nE)$ denote the Banach space of continuous $n$-homogeneous polynomials on $E$. Davie and Gamelin showed that the natural extension mapping from ${\cal P}(^nE)$ to ${\cal P}(^nE^{\ast\ast})$ is an isometry into the latter space. Here, we investigate when there is a natural isomorphism between ${\cal P}(^nE)^{\ast\ast}$ and ${\cal P}(^nE^{\ast\ast})$. Among other things, we show that if $E$ satisfies: \break (a) no spreading model built on a normalised weakly null sequence has a lower $q$-estimate for any $q < \infty,$ (b) $E^{\ast}$ has RNP, and (c) $E^{\ast}$ has the approximation property, then ${\cal P}(^nE)$ has RNP for every $n$. Moreover, if $E$ satisfies (a) and is such that $E^{\ast\ast}$ has both the RNP and the approximation property, then ${\cal P}(^nE)^{\ast\ast}$ and ${\cal P}(^nE^{\ast\ast})$ are isomorphic for every $n$. We also exhibit a quasi-reflexive Banach space $E$ for which ${\cal P}(^nE)^{\ast\ast}$ and ${\cal P}(^nE^{\ast\ast})$ are isomorphic for every $n$. 
Related work has been done recently by (i) M. Gonzalez, (ii) M. Valdivia, and (iii) J. Jaramillo, A. Prieto, and I. Zalduendo.

/pdf/q-reflexive-banach-spaces-23dt2dsd0x.pdf

$Q$-Reflexive Banach Spaces

Let E be an infinite dimensional complex Banach space. Let Hwu(e) be the space of entire complexvalued functions on E which are weakly uniformly continuous when restricted to any bounded subset of E, and let Hwsc(E) be the space of entire complex-valued functions on E which map weakly convergent sequences in E to convergent sequences. These and intermediate spaces are characterized in terms of compact holomorphic mappings and in terms of equicontinuity conditions on the spaces of n-homogeneous Taylor polynomial coefficients.

Weakly Uniformly Continuous and Weakly Sequentially Continuous Entire Functions

Compact polynomials and compact differentiable mappings between Banach spaces

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Compact polynomials and compact differentiable mappings between Banach spaces