let $E = C([0,1]),\,\,$ $K : E \to E, \,\, (Kf)(x) = \int_0^1K(x,y)f(y)dy$
also $\|K\| \leq a < 1$
I want to prove that there for $g \in E$ there exists a unique $f_g \in E$ that satisfies the following equation :
$f_g + Kf_g = g$
which is equivalent to showing that $T : E \to E,\,\,T(f) = g-Kf$ has a fixed point.
with what I have in hands I feel like there must be some theorem I'm missing.
any help will be greatly appreciated !