Let $X$ be a completely regular space. For every compact subset $K$ of X, define a seminorm $p_K: C(X)\to {\Bbb C}$ such that $p_K(f):=\sup_{x\in K}|f(x)|$. Then $\{p_K;K ~is ~compact \}$ is a collection of seminorms that makes $C(X)$ a locally convex space. Show that a net $\{f_i\}$ in $C(X)$ converges to $f$ iff $f_i\to f$ uniformly on compact subsets of $X$.
My attempt; first I let $\{f_i\}$ in $C(X)$ converges to $f$ then for every compact subset $K$ . $p_k(f-f_i) = \sup_{x\in K}|f(x)|\leq \sup_{x\in X}|f(x)| = p_X(f-f_i)$ which show that $f_i\to f$ uniformly on compact subsets of $X$.
for connverse direction I stuck. Please hint me. Thanks in advance.
