Most of the material on the page was from a rewrite I did a few month ago. I am busy so I do not have time to work on the article so I will just give a few short notes on your rewrite. To be honest I have some issues with your edits. I answered you question above and fixed your error in the introduction of the article. 

There are two methods to define the topology of a LCTVS space: using a family of seminorms and using a local basis of locally convex, balanced and absorbend sets. Both methods should be present in the article on equal terms. The family of seminorms is useful to view LCTVS as generalizations of normed vector spaces. The definition in terms of a local base is useful if you are coming from the more abstract topological vector space concept.

The definition section is now too long and detailed. I think the subject is specialized enough so that we can assume the reader is familiar with the basic concepts which are now explained in detail in the definition.

The article was not perfect before but you introduced errors and made the article less readable, at least from my perspective.