This is the opposite question to this one: [Example of locally convex space such that its weak and initial topology coincide](https://mathoverflow.net/q/156538/13356).


If we have a normed vector space $X$ than its norm topology and weak 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?
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> 


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.



Sufficient condition such that weak and initial topology coincide for a locally convex space
The following is mentioned in the book by [H.Schaefer](http://www.goodreads.com/book/show/2302578.Topological_Vector_Spaces) (IV, Exercise 6(a)), but with a little bit different formulation:


**Theorem.** A complete locally convex space $X$ over $\mathbb K$ ($\mathbb K=\mathbb R$ or $\mathbb C$) has weak topology (i.e. the topology of $X$ coincides with the weak topology on $X$ generated by all linear continuous functionals $f:X\to{\mathbb K}$) if and only if $X$ is isomorphic (as a locally convex space) to the direct product of a family of copies of $\mathbb K$:
$$
X\cong{\mathbb K}^{\mathfrak m}
$$
for some cardinality ${\mathfrak m}$. 


**Corollary**. If $X$ is complete and $X\not\cong{\mathbb K}^{\mathfrak m}$, then the topology of $X$ can't be weak. In particular, a Fréchet space $X$ has weak topology if and only if $X\cong{\mathbb K}^{\mathbb N}$.


