Then, S is [L.sup.0]-linearly homeomorphically embedded into [([S.sup.*.sub.s]).sup.*.sub.s] by the canonical mapping J : S [right arrow] J(S) [subset] [([S.sup.*.sub.s]).sup.*.sub.s] defined by J(x)(f) = f(x), [for all]x [member of] S and f [member of] [S.sup.*.sub.s], where [S.sup.*.sub.s] denotes the random conjugate space [S.sup.*.sub.c] endowed with its strong locally [L.sup.0]-convex topology.

Denote by [(E, P).sup.*.sub.[epsilon],[lambda]] the [L.sup.0](F, K)-module of continuous module homomorphisms from (F, [T.sub.[epsilon],[lambda]]) to ([L.sup.0](F, K), [T.sub.[epsilon],[lambda]]), called the random conjugate space of (F, P) under [T.sub.[epsilon],[lambda]]; denote by [(F, P).sup.*.sub.c] the [L.sup.0](F, K)-module of continuous module homomorphisms from (F, [T.sub.c]) to ([L.sup.0](F, K), [T.sub.c]), called the random conjugate space of (F, P) under [T.sub.c].

Based on the idea of randomizing functional space theory, a new approach to random functional analysis was initiated by Guo in [1-3]; in particular, the study of random normed modules and random inner product modules together with their random conjugate spaces was already the central theme in random functional analysis in [2, 3].

Specially, denote (B [S.sup.1][S.sup.2]), [parallel]* [parallel]) by (S*, [parallel]*[parallel]*) when ([S.sup.1], [[parallel]*[parallel].sub.1]) is a given RN module (S, [parallel] * [parallel]) over K with base ([OMEGA],F,P) and [S.sup.2] = [L.sup.0](F,K),then (S*, [parallel]* [parallel]*) is called the random conjugate space of (S, [parallel]* [parallel]).Let (S**, [parallel]* [parallel]**) be the random conjugate space of (S*, [parallel]* [parallel]*).

Since an RN module is often endowed with a natural topology, called the ([epsilon], [lambda]) topology, it is not a locally convex space in general and in particular the theory of classical conjugate spaces universally fails to serve the theory of RN modules.

and [([W.sup.1,p.sub.T]).sup.*] stands for the

conjugate space of [W.sup.1,p.sub.T].

[4,10,13] The

conjugate space of [L.sup.p(x)]([OMEGA]) is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where 1/p(x) + 1/[p.sup.o](x) = 1.

For a sequence space X its

conjugate space or a set of continuous linear functionals on X is denoted by [X.sup.f] and is defined by

Recall that Clifford algebra and Bott periodicity dictate that only [R.sup.4], [R.sup.8], and other real spaces [R.sup.n] with dimensions divisible by four have two equivalent

conjugate spaces, the specific mathematical property that accommodates both particle states and antiparticle states.