In Reply to: Week 6 correction/suggestion posted by Chris#2 on May 12, 2003 at 22:47:09:
>-To do Lemma 1.3 for HW#6, one needs to show that if the sequences of functions (f_n) converge uniformly to a function f, then f is also in the C(R/Z;C) space.
>-But right below this we are given a metric to work with, i.e. sup-norm.
>-Shouldn't the metric be shown before this lemma? Without the metric, how is one to show uniform convergence?
>---Chris#2
Uniform convergence is defined in Week 3 notes, page 6-7. That
definition does not require a metric, though Proposition 7 of
Week 3 does show that uniform convergence is equivalent to convergence in the sup norm metric (at least if one restricts one's attention to bounded functions).
So to prove Lemma 1(iii), you can either use the definition of uniform convergence directly, or you can use Proposition 7 from Week 3 to rewrite uniform convergence in terms of the sup norm metric. (Note that you already know that functions in C(R/Z;C) are bounded by Lemma 1(i)).
Terry