Assume that [OMEGA] is a completely regular Hausdorff space and ([OMEGA], Ba, [mu]) is complete finite measure space
. Let T : [L.sup.[phi]](X) [right arrow] Y be a ([[gamma].sub.[phi]], [[parallel]x[parallel].sub.Y])-continuous linear operator and m : Ba [right arrow] L(X, Y) be its representing measure.
Let ([OMEGA], [mu]) be a finite measure space
, and let M([OMEGA]) be the set of all measurable functions on [OMEGA] with values in [-[infinity], +[infinity]], [M.sup.+]([OMEGA]) be the subset of the nonnegative functions, and [M.sub.0]([OMEGA]) be the subset of the real valued functions.
Let (R, [summation], [mu]) be a finite measure space
. Let [phi] be an Orlicz function, i.e.
Let ([OMEGA], [summation], [mu]) be a finite measure space
, and consider an orthonormal sequence [([f.sub.i]).sub.i] of real functions in [L.sup.2]([mu]) and a sequence or real numbers [([a.sub.i]).sub.i].