Ah! This is a better question! This is why I asked what norm you were using. The canonical norm on

is the infinity norm (BECAUSE it makes it complete), but it is not complete as you mentioned with respect to the

-norm. So, to answer what I think is your question, which I think is asking does a sequence in

converge to an element of

always (since

is complete we know it converges to an element of

, but will this guy live in

, the answer is no. We enlarged our space precisely because this isn't true.

Is this what you were looking for?