Operator representation theorems
Edward O. Thorp, Robert J. Whitley · Illinois Journal of Mathematics · 1965
We consider representations of bounded, compact and weakly compact linear operators from a Banach space to a space BC(S), where S is an arbitrary topological space and BC(S) is the space of bounded continuous scalar-valued functions on S with the sup norm.With the use of our theorems, one can quickly and easily deduce numerous operator representation theorems, many of which are new.For example, taking as domain space a space with a well- known conjugate space and range space as either co or m, one fills in quite a few blanks in Tables VI A, B and C in [4].Our proofs for range BC(S), S arbitrary, are appreciably simpler than those found in the literature for range C(S), S a compact Hausdorff space.The spaces of bounded, compact and weakly compact linear maps from a B-space X to a B-space Y will be denoted, respectively, by B[X, Y], K[X, Y] and W[X, Y].Unexplained terminology and notation will be found in [4].Phillips [9] represented the general bounded operator from X to B(S).Gelfand [5] represented the general bounded and compact operator from a B-space X to C[0, 1] while Sirvint [11], [12] represented the general weakly compact operator.More recently, Bartle [1, Theorem 10.2] represented these three types of operators mapping X into the space C(S) of continuous functions on a compact Hausdor space S.However, Bartle's theorem is