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
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)