Exercise 2, § 8

Let K be a field, K' a subfield of K, V a right vector space over K, V' a K'-structure on V and R' the canonical image of the dual V'* in the dual V* of V (no. 4). For R' to be a generating system of V*, it is necessary and sufficient that V be finite-dimensional or that K be a finite-dimensional right vector K'-space (use Exercise 25 of § 7).