Differentiable norms in Banach spaces

Abstract

413connecting the fixed points of I\ (4) If Ox and 02 are disjoint simply connected domains invariant under a loxodromic T f the corresponding arcs, as in (3), divide S into two Jordan regions, one or the other of which must contain any domain disjoint from 0\ and 0%. (5) If 0 is a simply connected domain invariant under an elliptic T t then O must contain a fixed point of T.The examples are elaborations of the ideas in L. R, Ford, Automorphic functions y 2nd éd., Chelsea, 1951, pp. 55-59.

Smooth Locally Convex Spaces

“…2.8 generalises the result of Kadec [16] and Restrepo [15] to locally convex spaces. Stronger versions of the Kadec-Restrepo result have been obtained in Banach spaces by Leach and Whitfield [7] and Leduc [8].…”

“…It is not known if the converse is true, although some partial converses are known. For example, combining the results of Leach and Whitfield [7] and Restrepo [15], if F is a separable, D^-smooth Banach space, then F is strongly £)¿.-smooth. See also [9].…”

Geometric properties of the norm and basic sequences in Banach spaces

“…The equivalence 1° '^ 3° can be also proved by using characterizations of (F) in terms of "strongly smooth" p o i n t s , and of (S), due to Smulian [7S], [77] (see also [ 7 / ] ) ; in the p a r t i c u l a r case of reflexive spaces, essentially t h i s l a t t e r equivalence has been obtained, with a different method, in [ 5 ] . In [74], Proof of Theorem 3, i t has been shown, with the argument of the above proof of the implication 3° ' " I 0 , that (S) A (KK^) °* (F). In [?6] we have proved some results on the open problem whether the second conjugate E** of a non-reflexive Banach space E is non-smooth, showing that for a large class of spaces t h i s is indeed the case.…”

Differentiable Functions and Rough Norms on Banach Spaces

“…Bonic and Frampton [1] removed the continuity condition. Restrepo [4], [5] and Kadec [2] proved that if X* has density character greater than that of X (dens A'*>densA'), there does not exist a Frechet-smooth norm for X. Leduc [7] has proved that if dens X*>dens X, then there does not exist any continuouslv Frechet differentiable function on X with bounded nonempty support. His proof is simpler than the proof of the stronger result in this paper.…”