Editing Topological vector space (section)

===Non-normed spaces===

There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of [[holomorphic function]]s on an open domain, spaces of [[infinitely differentiable function]]s, the [[Schwartz space]]s, and spaces of [[test function]]s and the spaces of [[Distribution (mathematics)|distributions]] on them.{{sfn|Rudin|1991|p=4-5 §1.3}} These are all examples of [[Montel space]]s. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by [[Kolmogorov's normability criterion]]. 

A [[topological field]] is a topological vector space over each of its [[Field extension|subfields]].