Vector Spaces over General Fields

Abstract

The first three paragraphs are concerned with the elementary and purely algebraic properties of vector spaces E over a general commutative field. In § 7 the lattice V(E) of linear subspaces of E is studied, and § 8 deals with linear mappings from one vector space into another, and their representation by infinite matrices. The problem of the equivalence of these mappings is completely solved. The algebraic dual space E* of all linear functionals on E is the theme of § 9. The lattice $\bar V(E*)\$ of algebraically closed subspaces of E* turns out to be dually isomorphic with V(E). The end of § 9 is concerned with the most important elementary properties of tensor products of vector spaces.