Question from a topology textbook regarding the uniform topology 
I am able to prove that T maps continuous functions to continuous functions but I am lost when it comes to proving T is linear. I am also unsure how to solve b) and c). Thanks for any help you can offer! 

(a) To check linearity, just use that integrals are linear:
$$ T(cf+g)(x) = \int K(x,y)(cf(y)+g(y))\ dy =  \cdots $$
(b) To check continuity in the uniform topology, take a sequence of functions $f_n$ which converge to $f$ in the sup norm. Then consider $$\|Tf_n - Tf\|_\infty = \sup_x\bigg(\int K(x,y)f_n(y)\ dy - \int K(x,y)f(y)\ dy\bigg).$$ Can you see how to use convergence of $f_n\to f$ in the sup norm to show continuity?
(c) There are several ways to attack this. One way is to construct a sequence of Fredholm operators converging to $T$ in the operator norm. There are plenty of proofs out there using this method; google "Hilbert-Schmidt operators are compact" for some of them. (The gist is approximating $K$ by simple functions.) Another way is to directly show it by exhibiting Cauchy subsequences of images of sequences in the unit ball.
