Proposition 1

(i) For a left vector space over K to be of finite dimension n, it is necessary and sufficient that it be isomorphic to Ks^n.*

(ii) For two vector spaces Ks^m and K_s^n to be isomorphic (m and n integers $\geq 0$), it is necessary and sufficient that m = n.*

(iii) In a vector space E of finite dimension n, every generating system has at least n elements; a generating system of E with n elements is a basis of E.

(iv) In a vector space E of finite dimension n, every free subset has at most n elements; a free subset with n elements is a basis of E.