# Thread: Banach algebras question 1

1. ## Banach algebras question 1

I've questions about two steps in the proof of the following theorem:

Red shape: From where we know that this functional is continuous?
Blue shape: From where we know that completion of A is a Sub-norm algebra?

PS. Here is link to the original paper: http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf

2. Red: If $f(x) = x(p_x)$ then $|f(x)|\leqslant \|f\|_{\text{sup}}$. Thus the linear functional f is bounded and hence continuous for the sup norm.

Blue: The completion of any normed algebra is a normed algebra for that norm.