Hi and happy new year,

Here is a problem in metric spaces. And don't worry; it's not a homework. It's just that my exam is in 2 days so I have to be prepared. Here we are:

In the metric space (C[a,b] , dinf) consider

Ak = {g:[a,b]-->R | |g(t1) - g(t2)|<or = k|t1 - t2| for all t1,t2 in [a,b]}

Bk = {f:[a,b]-->R | f is differentiable and |f'(t)|< or = k for all t in [a,b]}

Show that:

1) Ak is closed.

2) Ak = the closure of Bk.

3) A = union of all Ak, k>0 is not closed.

4) The closure of A = C[a,b]

I solved (1) that Ak is closed and in (2) I was able to prove that the closure of Bk is a subset of Ak (which results from Lagrange and (1)). The problem now is to prove that Ak is a subset of the closure of Bk and to prove (3) and (4).

Thanks a lot