Hm... I guess it would help if I looked up what local convexity means. I think I've looked it up before and if I remember correctly, it means the topology on the space is induced by a bunch of seminorms.
@MattN A linear functional $\varphi$ is continuous if and only if the seminorm $x \mapsto |\varphi(x)|$ is continuous. Since the weak topology is the initial topology induced by the linear functionals it is also the initial topology induced by those seminorms, so it is locally convex.
@tb I see. I didn't know the thing about the seminorms. I think I should look at some more basic stuff first because I even struggle to understand everything your 'prediction' : )
@MattN I think what you should understand right now is that on an infinite-dimensional space the weak topology is never metrizable (it's not even first-countable), hence it is weaker than the topology you start with in most situations where you don't already know that you start with the weak topology.
@MattN Do you know that a Hausdorff locally convex space is metrizable if and only if it is first countable? If it is first countable, there are countably many seminorms $\{|\cdot|_n\}_{n=1}^\infty$ inducing the topology. Then the metric $d(x,y) = \sum_{n=1}^\infty 2^{-n} \min{\{|x-y|_n,1\}}$ is compatible with the topology.
