This is the opposite question to this one: Example of locally convex space such that its weak and initial topology coincide.

If we have a normed vector space $X$ than its norm topology and initial topology coincide if and only if $X$ is finite dimensional. Now I'm interested in the general case of $X$ being just locally convex.

If $X$ is a locally convex space (which is not normed) what are sufficient conditions such that the weak topology and the initial topology of $X$ do not coincide?

I think that there must be at least one quite general condition, since I think that it is rather rare that we have a locally convex space fulfilling the condition that its weak and initial topology coincide.