Prove Banach space X endowed with weak topology is locally convex.
To prove it has local convex base is easy since we can write down the neighborhood explicitly
To question is do we need to prove the space X with weak topology is topological vector space.I don't know how to prove this since the topology is no longer norm topology?
I try to prove it by definition that is prove addition map $+:X\times X \to X$ and scalar are continuous.(it seems not very hard by definition correct?)
Best Answer
In my opinion yes, since you know have a new topology, you should prove again that the new topology leads to a topological vector space.
Your attempt in the comments sounds correct to me so far. Of course, one should also show that the multiplication by a scalar is continuous.