You're right that it should be ![||f_n||_\infty = \sup_{x\in [0,1]} |f_n(x)|](https://kurser.math.su.se/filter/tex/pix.php/4c72297b15c53c009351127fb1263692.png "||f_n||_\infty = \sup_{x\in [0,1]} |f_n(x)|")
. Definition 3.16 in Rudin defines 
, it doesn't have any different meaning in this context.
. Definition 3.16 in Rudin defines , it doesn't have any different meaning in this context.. Definition 3.16 in Rudin defines , it doesn't have any different meaning in this context.