I have this question to solve, but I have no idea how to solve:

a) Let $X$ a normed space and $\{ x\_n \}\_{n \in \mathbb{N}} \subset X$ a sequence with $\sum\_{n=1}^\infty | x'(x\_n) | < \infty$ for all $x' \in X'$. Show that $\sup\_{\|x'\| \leq 1} \sum\_{n=1}^\infty | x'(x\_n) | < \infty$.

b) Let $K \subset C([a,b])$ be $K = \{ f \in C([a,b]) : \int\_{a}^{(a+b)/2} f(x) \mathrm{d}x = \int\_{(a+b)/2}^{a} f(x) \mathrm{d}x \}$. Show that $K$ is a closed vector subspace and find $x' \in C([a,b])'$ so that $\ker (x')=K$.

Thanks for any help :)