Examples
All normed vector spaces, and therefore all Banach spaces and Hilbert spaces, are examples of topological vector spaces.
However, 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 functions on an open domain, spaces of infinitely differentiable functions, the Schwartz spaces, and spaces of test functions and the spaces of distributions on them. These are all examples of Montel spaces. On the other hand, infinite dimensional Montel spaces are never normable.
A topological field is a topological vector space over each of its subfields.
Read more about this topic: Topological Vector Space
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