Trees in Renorming Theory

Smooth Norms and Approximation in Banach Spaces of the Type 𝒞(k)

“…Formalizing a definition which appears implicitly in [6,7], we shall say that a mapping T : C(K) → c 0 (K × Γ) is a (non-linear) Talagrand operator of class C m if…”

“…The analysis in that paper of compact spaces constructed using trees suggested that for a compact space K the existence of an equivalent norm on C(K) which is of class C 1 (except at 0 of course) might imply the existence of such a norm which is of class C ∞ . Certainly, this is what happens with norms constructed using linear Talagrand operators as in [5,6,7]. The other important (and older) method of obtaining C 1 norms is to construct a norm with locally uniformly rotund dual norm.…”

Spaces of functions with countably many discontinuities

“…It is natural therefore to hope for good results when we look at these compact spaces in the context of renorming theory. In fact, however, as Todorcevic [15] has recently observed, there is a scattered Rosenthal compactification K of a tree space such that C(K) has no LUR renorming, [6]. Now that space K is non-separable and, as other recent work of Todorcevic [16] has shown, it is only from separable Rosenthal compacta that we should expect really good behaviour.…”

“…As in [6] and [7], this follows from Banach's Contraction Mapping Theorem applied in a suitable function space. To describe it we take as domain for our functions the set…”

“…As in [6] and [7] we shall employ a method of recursive definitions, combined with the following result, which we refer to as Deville's Lemma. Let us mention that an approach based on countably decomposition in the spirit of [11] is also possible.…”