Completion of normed linear spaces

Abstract

2. Notation and definitions. Let F be the field of real numbers or the field of complex numbers. The completion A(X) of a normed F-linear space X is usually constructed as the quotient A(X) -C(X)/co(X) of the normed linear space C(X) of all Cauchy sequences [xl, X2, * I in X (with norm I [xI, X2, ** =sup4 x|j) by the subspace co(X) of sequences tending to zero. It is well known that the passage X-A(X) is a functor from the category N of normed linear spaces and contractions (continuous linear maps f: X-+ Y such that IfI A(X) is just the composition c(X) =m(X)l(X), where 1(X): X-+C(X) is defined by 1(X) (x) = [x, x, ] and m(X) is the quotient projection C(X)-4A(X). It is useful to note that X.-C(X) and Xs-+co(X) are functors from N to N (if f: X--+Y, then C(f): C(X)-+C(Y) is the map C(f)([XI, X2, * * ]) = [f(x),f(x2), * * * ]) and co is a subfunctor of C. We will denote by n: co--C the inclusion. Note also that I defined above is a natural transformation from the identity functor on N to the functor C and m is a natural transformation from the functor C to the functor A. DEFINITION. A (finite or infinite) sequence in N' (or B)