The adjoint of a bilinear operation
Richard F. Arens · Proceedings of the American Mathematical Society · 1951
m': YXX->Z by ml(y, x) =m(x, y).Since m1*** is an extension of m', the operation m'***1 is another extension of m.The question of commutativity mentioned above is included in the question: Is mt***t -m***f jf this is true let us say that m is regular.We shall show below that this is not always true.(In fact, we even show that if m is a commutative associative normed-algebra-operation of multiplication, then m*** need not be commutative.)An important special case of an m which satisfies 1.0-1.3 is bounded bilinear functional, where this word is used, as often, to denote that m(x, y) is a number, that is, element of K.When m satisfies 1.0-1.3 it is possible to characterize m*** in terms of weak convergence (see §3) as follows: an extension n of m to X~~X Y~~ with values in Z coincides with m*** if and only if n(a, ß) is weakly continuous in a when ß is fixed, and n(a, ß) is weakly continuous in ß when a is any fixed element of X (where X is regarded as embedded in X~~ for the moment).Mutatis mutandis, such a characterization holds for m'***' also.Thus the regularity of m can be regarded as a certain double weak continuity of m*** and/or of m'***'.Note: In reading this paper, one does well to keep in mind that XX Y^Z, Z-XX-* Y-, Y-XZ-^ X-, X-X Y--+Z-.2. Reduction to functionals.Let Z and W be normed linear spaces, and h be a bounded linear operation of Z into W, in symbols,