We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write

(1.1)

If

(1.2)

as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,

(1.3)

We use the notations of [2], particularly for operators on classes of structures; in addition, P*r(K) (respectively, Pr(K)) denotes the class of all reduced products of families of structures in K (those with respect to a proper dual ideal, respectively).