# Total set

From Encyclopedia of Mathematics

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 .

#### Comments

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.

#### References

[a1] | S. Rolewicz, "Metric linear spaces" , Reidel (1985) pp. 44 |

[a2] | G. Köthe, "Topological vector spaces" , 1 , Springer (1969) pp. 132, 247ff |

**How to Cite This Entry:**

Total set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Total_set&oldid=14064

This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article