# Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space.

Consider the properties:

a) $C$ is closed.

b) $C$ is weakly closed.

c) $C$ is weakly sequentially closed.

d) $C$ is sequentially closed.

Then a) $\Leftrightarrow$ b) $\Rightarrow$ c) $\Rightarrow$ d).

Questions:

Does c) imply b)?

Does d) imply c)?

(In any normed vector space, d) implies a), so all notions are equivalent for convex subsets of normed vector spaces.)

• @AlexR: is it ok now? – user66081 Dec 6 '14 at 22:13
• Now everything is clear. Thanks for taking the time to clarify :) – AlexR Dec 6 '14 at 22:29