On my textbook (Frigyes Riesz, Bela Sz.-Nagy, functional analysis) there is the following theorem.
Theorem. A necessary and sufficient condition that the functional A, given in a set E of the space C, be extendable to the entire space C so as to define there a linear functional of norm $\leq M$ is that $$\left|\sum_{1}^n c_k A f_k\right|\leq M \left|\left|\sum_{1}^n c_k f_k\right|\right|$$ for every linear combination of elements of E.
The theorem's proof is not very clear for me. Is there any good reference (online, if possible)?
Thanks!