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

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

### Citations:

Zbl 0731.00013; Zbl 0084.09805; Zbl 0312.46031
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