Random linear functionals

Abstract

r.l.f.'s are useful and will be discussed in ?1 below. If T is a topological linear space let T' be the topological dual space of all continuous linear functions on T. If T'=S, then a random linear functional over S defines a weak distribution on Tin the sense of I. E. Segal [23]. See the discussion of semimeasures in ?1 below for the relevant construction. A central question about an r.l.f. (Sa, P), supposing S is a topological linear space, is whether P gives outer measure 1 to S'. Then we call the r.l.f.,canonical, and as is well known P can be restricted to S' suitably (cf. 2.3 below). For simplicity, suppose that for each s E S, f x(S)2 dP(x) < 0. Let C(s)(x) = x(s). Then C is a linear operator from S into the Hilbert space L2(Sa, P). One seeks conditions on this operator to make the r.l.f. canonical. If S itself is a Hilbert space, the fundamental theorem of Sazonov [22] and associated results due to Minlos [20] and others assert that it is sufficient for C to be a Hilbert-Schmidt operator, and that this result is the best possible. If S is a Banach space, it is sufficient for C to be nuclear in the sense of Grothendieck [15]; see 8.5 below. The conjecture that it would suffice for C*C to be nuclear, as if S is Hilbert, is not true even in the Gaussian (mean 0) case, where f eix(s) dP(x) =e- Q(s), Q being a nonnegative definite quadratic form on the Banach space S; see the end of ?6 below.