Posted by oo Un,g) sin(n + m1r) 2(n + m1r) forgE Ka = {O,g) = 0, and Un} thus converges weakly to f = 0. 3. Convergence of Sequences of Functions 33 We have seen that, in a given lJ' space, strong convergence implies weak convergence. It also turns out that we may compare convergence in different lJ' spaces using the following proposition. 1. (X). 2 belongs to LP1 , and strong converyence in £P2 implies strong converyence in IJ'1 • Proof.
5) This is the Cauchy condition for convergence. 5) is also sufficient for convergence. This is stated more precisely in the following theorem. 1. 5) holds. 3) holds. 1 holds for LP spaces is referred to by saying that LP spaces are complete. 1 enables us to prove the convergence of series by the use of a comparison series. Suppose we have a sequence {gn} C LP and we know the series of norms IIDniiLP is convergent, that is, L IIYniiLP < 00. 7) is also strongly convergent and that its sum is an element of LP.