Interpolation Spaces by Complex Methods

where the infimum is taken over all pairs x^ÇzXj satisfying (2.1). One easily checks that (2.2) gives a norm on X 0 + X i when X0 and X% are compatible. Moreover, when X0+Xi is equipped with this norm, it becomes a Banach space (cf. [3]). Let Cfc denote the set of complex valued functions of the complex variable f = ̂ +irf which are continuous on bounded subsets of S, where fl is the strip 0 <£ < 1 in the f = £+«7 plane. Let (B be the set of those functions in Ct which are holomorphic in 0 and nonvanishing in Ô. For p£(5t the space 3C(X0, -XV, p) will consist of those functions /(f) with values in X0+Xi such that (a) ƒ($") is continuous on bounded subsets of S, (b) /(f) is holomorphic in 0, (c) f(j+iv)€:Xj,j = 0, 1,77 real, and