FortopologicalspacesXandY,letC(X,Y)denotethecollectionofallcontinuousfunctionsfromXtoY.Wewilldistinguishtwospecialcollections: C(X)willbeusedtodenoteC(X,R)andC*(X)willdenotethesetofallbounded functionsfromC(X).Wecandefineaddition,multiplicationandscalarmultiplication offunctionsinC(X)pointwise:
(f+g)(x)=f(x)+ g(x),
(f·g)(x)=f(x)·g(x),
(a·f)(x) =a·f(x),fora?R.
1. If fandgbelongtoC(X),thensodo f +g,
f·ganda·f,fora?R.
If,inaddition,fandgarebounded,thensoaref+g,f·ganda·f
2.C(X)andC*(X)arealgebrasovertherealnumbers. (Consultanybookonabstractalgebraforthedefinitionofan algebra.)
3.C*(X)isa normedlinear space (2J) with the operations ofadditionand scalarmultiplicationgivenaboveandthenorm || f||=sup x?X lf(x)l.