is finite and there exists anxo E H such that 111(xo)11 = IIIIIH.

(b) A sequence of functions Ik : H ~ Rm is said to converge uniformly on H to a function I : H ~ R m if and only if for every c > 0 there is an N E N such that k ~ N and x E H imply Ilik(x) - I (x) II < c.

Show that Ilik - IIIH ~ 0 as k ~ 00 if and only if ik ~ I uniformly on H as k ~ 00.

(c) Prove that a sequence of functions ik converges uniformly on H if and only if for every c > 0 there is an N E N such that k,j ~ N implies Ilik -hlIH < c.