Proposition 5

Let V be a right vector space over K, V' a K'-structure on V and W a vector sub-K-space of V. For W to be rational over K', it is necessary and sufficient that there exist a set H \subset V^* of rational linear forms over K' such that W is the orthogonal of H in V (§ 2, no. 4).

Let H be a subset of V* whose elements are rational linear forms over K'. For all $x^* \in H$, the kernel of $x^*$ is a vector sub-K-space of V, rational over K' (no. 3, Corollary 2 to Proposition 3); the intersection of these kernels is therefore also a vector sub-K-space of V, rational over K' (no. 2, Corollary 1 to Proposition 2).

Conversely, let W be a vector sub-K-space of V rational over K', so that W is identified with $W' \otimes_{K'} K$, where $W' = W \cap V'$ (no. 2, Proposition 2). For a linear form ${x'}^* \in {V'}^*$ to be zero on W', it is necessary and sufficient that the linear form $x^* \in V^*$ which corresponds to it under (1) be zero on W, for by no. 3, Corollary I to Proposition 3,

Let H' be the orthogonal of W' in V'*; we know (§ 7, no. 5, Theorem 7) that W' is the orthogonal of H' in V'; if H is the image of H' in V* under the mapping (1), it follows from the above that W is the orthogonal of H in V, taking account of no. 7, Corollary to Proposition 14.