Proposition 2

Let V be a right vector space over K and V' a K'-structure on V. The mapping W' \mapsto W'.K is a bijection of the set of vector sub-K'-spaces of V' onto the set of vector sub-K-spaces of V which are rational over K' and the inverse bijection is W \mapsto W \cap V'.

Clearly the bijection $\lambda^{-1}: V \to V' \otimes_{\mathbf{K'}} \mathbf{K}$, the inverse of the bijection $\lambda$ defined in no. 1, Proposition 1, maps every vector sub-K'-space W' of V' onto its image under the canonical injection $x' \mapsto x' \otimes 1$ and W'.K onto W' $\otimes_{\mathbf{K'}} \mathbf{K}$; the assertions of Proposition 2 are thus consequences of Definition 2 and § 7, no. 9, Proposition 19.