I am currently reading about C* from the following notes ( http://www.math.uvic.ca/faculty/putnam/ln/C%2A-algebras.pdf ).
In the proof of GNS construction theorem 1.12.4 page 50 there is something I don't understand in particular 3 that it defines a bounded linear operator on $H_{\phi}$ if we have the following
$\|\pi_{\phi}(a)(b + N_{\phi})\|^2 = \langle[ab],[ab]\rangle = \phi((ab)^{*}ab) = \phi(b^{*}a^{*}ab) \leq ||a||^{2}||b + N_{\phi}||^{2}$
Is the reason it is continous is because it is bounded for every choice of ab ? can someone explain as to why its continous in details ? since this is the only part I don't understand in the proof.
The operator you are defining is $\pi_\phi(a)$ for a fixed $a$. The inequality you posted shows that $\pi_\phi(a)$ is bounded on the subspace of elements of the form $b+N_\phi$, for all $b$. A bounded operator is continuous.