Uniformity in linear spaces

The first chapter of this paper concerns itself with questions of uniform boundedness of sets of points in a Banach space and sets of functionals on a Banach space, as well as with a group of closely related resonance theorems. A well known example coming under this heading is the theorem of Toeplitz [36] t stating that supm Z I= a,.I < oo providing 77m =Zn-, -amntn converges whenever S, does. Another is the theorem of Hahn [12 ] stating that if an arbitrary continuous function has the partial sums of its Fourier expansion, with respect to an orthonormal sequence of bounded functions (n, essentially bounded, then the sequence f ZX= 1c^,=(x)co,(t) I dt is also essentially bounded. Still another is the theorem stating that if the adjoint of an everywhere defined transformation between Banach spaces is everywhere defined, then the transformation is continuous. This was proved, at least for Hilbert space, by von Neumann [21 ], Stone [34 ], Tamarkin ( [34], p. iv), and Stone and Tamarkin [35 ] and is probably not usually thought of as a theorem on uniform boundedness. Most of the results w-e have in the first chapter are new, others have been proved only in special cases, and some are well known but the proofs heretofore given have been different. Previous methods for discussing questions of uniform boundedness divide themselves into three groups (i) those associated with the names of Lebesgue [18], Banach [2], Hahn [11], and Hildebrandt [13 ]; (ii) those characterized by the elegant and direct use of the Baire category theorem as in the works of Banach [1], Saks [31], Saks and Tamarkin [32], and others; and (iii) those employed in a recent theorem of Gelfand [9] the proof of which is closely related to that of the category theorem. The second chapter of this paper is concerned with cases where a weak limiting process implies a strong one. These cases seem to be rather rare but probably more will come to light in the future. The theorem quoted above, stating that the existence of the adjoint implies the continuity of the function, belongs to the class of questions discussed in Chapter II as well as those discussed in Chapter I, while its analogue (Theorem 42) in the Boolean ring of