Let F be a field, X an infinite set, and V the free left F-module (vector space) on the set X. Let F^x be the set of all functions. ∫: X → F.

(a) F^x is a (right) vector space over F (with ( ∫ + g)(x) = ∫(x) + g(x) and (∫r)(x) = r∫(x)).

(b) There is a vector-space isomorphism V* ≅ F^x.

(c) dim_F F^X = |F|^|X| (Exercise 8.10).

(d) dim_F V* > dim_F V

Data from Exercise 8.10

If S is a multiplicative subset of a commutative ring R and I is an ideal of R, then s^-1(Rad I) = Rad (S^-1I) in the ring s^-1R.