Total set

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A set of linear functionals on a vector space separating the points of , that is, such that for any non-zero vector there is an with .


A total set in the sense above is also, and more precisely, called a total set of linear functions, [a1].

More generally, a set , where is a topological vector space, is called a total set or fundamental set if the linear span of is dense in . If , the algebraic dual of , is given the weak topology (so that , if is the base field and is an (algebraic) basis for ), the two definitions for a set agree.


[a1] S. Rolewicz, "Metric linear spaces" , Reidel (1985) pp. 44
[a2] G. Köthe, "Topological vector spaces" , 1 , Springer (1969) pp. 132, 247ff
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Total set. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article