A selection theorem

It is shown that under certain assumptions the existence of a continuous selection on every compact subset of a metric space implies the existence of a continuous selection on the whole space. An application of this result to a problem concerning the extension of compact operators is given. Research supported in part by the National Science Foundation, U.S.A. (NSF-GP-378).     

On operators which attain their norm

The following problem is considered. LetX andY be Banach spaces. Are those operators fromX toY which attain their norm on the unit cell ofX, norm dense in the space of all operators fromX toY? It is proved that this is always the case ifX is reflexive. In general the answer is negative and it depends on some convexity and smoothness properties of the unit cells inX andY. As an application a refinement of the Krein-Milman theorem and Mazur’s theorem concerning the density of smooth points, in the case of weakly compact sets in a separable space, is obtained.     

Extensions of lipschitz maps into Banach spaces

It is proved that ifY ⊂X are metric spaces withY havingn≧2 points then any mapf fromY into a Banach spaceZ can be extended to a map fromX intoZ so that wherec is an absolute constant. A related result is obtained for the case whereX is assumed to be a finite-dimensional normed space andY is an arbitrary subset ofX. Supported in part by US-Israel Binational Science Foundation and by NSF MCS-7903042. Supported in part by NSF MCS-8102714.     

An extension of a result of ribe

It is proved that for every 1≦p<∞, 1≦q<∞ and for every sequence {p _n}, 1≦p _n<∞,p _n→p, the spaceX=(Σ⊕l _{p n}) _q (resp.U=(Σ⊕L _{p n}(0, 1)) _q ) is uniformly homeomorphic toX⊕l _p (resp.U⊕L _p(0, 1)). This extends Ribe’s result from the casep=1 to generalp<∞ and thus provides examples of uniformly convex, uniformly homeomorphic Banach spaces which are not Lipschitz equivalent.     

Boundaries and generation of convex sets

We introduce a notion which is intermediate between that of taking thew*-closed convex hull of a set and taking the norm closed convex hull of this set. This notion helps to streamline the proof (given in [FLP]) of the famous result of James in the separable case. More importantly, it leads to stronger results in the same direction. For example:

On james’s paper “separable conjugate spaces”

For every separable Banach spaceX there is a Banach spaceY with a separable dual such thatY ⊕X* ≈Y**. There is also a separable spaceZ so thatZ**/JZ is isomorphic toX.     

On subspaces of Banach spaces without quasicomplements

It is proved that for uncountableΓ,c ₀(Γ) has no quasicomplement inm(Γ). Research supported by the National Science Foundation, U.S.A.     

Laminar two-phase boundary layers in natural convection film boiling of subcooled liquids

Zusammenfassung Es wird die zweiphasige Grenzschichtströmung behandelt, die in unterkühlter Flüssigkeit entsteht, wenn die wärmeabgebende (vertikale) Wand eine Temperatur oberhalb des Leidenfrost-Punktes hat. Unter Einschluss des Impulstransportes an der Phasengrenzfläche werden asymptotische Lösungen ermittelt, die für sehr geringe und sehr hohe Flüssigkeitsunterkühlung gelten. Von den beiden Grenzfällen ausgehend lässt sich eine Interpolationsformel angeben, welche mit Wärmeübergangsmessungen verglichen wird, die das thermische Verhalten des organisch gekühlten Reaktors simulieren. In erster Näherung stimmen Rechnung und Experiment überein, wenn man die Streuung der Messwerte und vereinfachende Rechnungsannahmen berücksichtigt.     

Notes on Klee’s paper “Polyhedral sections of convex bodies”

In sections 2 and 3 two methods for proving the non existence of certain universal Banach spaces, are presented. In section 4 it is proved that every infinite-dimensional conjugate Banach space has a two-dimensional subspace whose unit cell is not a polygon. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18, through the European Office of Aerospace Research (OAR) United States Air Force.