On certain classes of Baire-1 functions with applications to Banach space theory. (English) Zbl 0762.46006

Haydon, R.; Odell, E.; Rosenthal, H.

Functional analysis, Proc. Semin., Austin/TX (USA) 1987-89, Lect. Notes Math. 1470, 1-35 (1991).

The problem whether an infinite dimensional Banach Space \(X\) contains an (1) infinite dimensional reflexive subspace or an isomorph of (2) \(c_0\) or (3) \(\ell_1\) is one important motivation of the presented approach. For a separable \(X\), properties close to (1) – (3) can be characterized in terms of behavior of certain type of continuous functions on the dual ball \(K\) in \(X^*\), which is a compact metric space under the weak\(^*\) topology.
Namely,
(1): \(X\) is reflexive iff \(X^{**}\subset C(K)\);
(2): \(c_ 0\hookrightarrow X\) iff \(X^{**}\setminus B_ 1(K)\neq \emptyset\) [C. Bessaga and A. Pełczyński, Stud. Math. 17, 151–164 (1958; Zbl 0084.09805)];
(3): \(\ell_ 1 \hookrightarrow X\) iff \([X^{**}\cap DBSC(K)] \setminus C(K)\neq \emptyset\). [E. Odell and H. Rosenthal, Isr. J. Math. 20, 375–384 (1975; Zbl 0312.46031)].
Here, \(B_1(K)\) is the class of bounded Baire-1 functions on \(K\) (pointwise limits of uniformly bounded pointwise convergent sequences of continuous functions on \(K\)), and \(DBSC(K)\) consists precisely of “differences of bounded semicontinuous functions on \(K\)”, i.e., each \(F\in DBSC(K)\) admits a pointwise convergent representation \(F=\sum_k g_k\), \(g_k\in C(K)\) such that \(\sup_K \sum_k | g_k| \leq C<\infty\). The least constant \(C\), denoted by \(| F|_D\), turns \(DBSC(K)\) into a Banach space. Moreover, \(C(K)\subset DBSC(K)\subset B_1(K)\).
In the paper, the case \(X^{**}\cap [B_1(K)\setminus DBSC(K)]\neq \emptyset\) is studied in detail with the help of various (proper, and distinct among themselves) subclasses of \(B_1(K)\), containing \(DBSC(K)\) as a proper subspace. Many results are given in the context of arbitrary metric compacta \(K\). Also, an extensive bibliography on related topics is included.
[For the entire collection see Zbl 0731.00013.]

Reviewer: J. Szulga (Auburn)

MSC:

46B20	Geometry and structure of normed linear spaces
46A25	Reflexivity and semi-reflexivity
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
26A45	Functions of bounded variation, generalizations

Keywords:

class of bounded Baire-1 functions; differences of bounded semicontinuous functions; metric compacta; isomorphic structure; extreme points; variation of functions; semicontinuous

Citations:

Zbl 0731.00013; Zbl 0084.09805; Zbl 0312.46031