Closure, Convexity, and Linearity in Banach Spaces

Abstract

The work of Banach [1],2 Mazur [2], and others has emphasized the fundamental role played by convex functions and sets in the study of Banach spaces and (continuous) linear transformations between them. It is natural to suspect that similar considerations underlie the theory of general linear transformations, in which closure rather than the restricted notion of continuity is predominant. In this paper, therefore, we study linear transformations T from a Banach space B1 , to a Banach space B2, where T denotes a single-valued, additive and homogeneous operation with linear domain DT in B1, and linear range RT in B2. Our definition of linearity is purely algebraic and will necessarily imply continuity and total definition (DT = B1) cnly in the case of linear functionals. With these exceptions our terminology conforms to that of Banach. When T is linear the expression K(x) = 11 Tx 11, xe DT is a convex function, and E[K(x) _ a], a > 0, is a convex set, where E[K(x) ? a] denotes the set of points x e DK = DT such that K(x) ? a. An examination of the connections between the closure of T, the closure of E[K(x) < a], and the structure of convex K's yields a number of theorems with interesting applications, including as special cases theorems of Gelfand [3] and Bosanquet and Kestelmann [4] of the Banach-Steinhaus type (cf. Zygmund [5], pp. 97-100). Upon specializing B1 and B2 to Hilbert space we obtain a characterization of the domain of a selfadjoint transformation. A by-product of the analysis is a surprisingly simple proof of Wintner's theorem [6, ?108]: every definite symmetric transformation possesses a self-adjoint extension.