Proposition 9

Let L be a field and E a subring of End$_{\mathbf{Z}}(L)$ that contains the mapping $\boldsymbol{\gamma }_a:x\mapsto ax$ for every $a\in L$. Denote by K the set of elements $a$ of L such that $u(xa) =u(x)a$ for every $x$ in L and every $u$ in E; it is a subfield of L.

Let V be a finite-dimensional linear subspace of the right K-vector space L, and let $h$ be a K-linear mapping from V to L. There exists an element of E that coincides with $h$ on V.

We view L as a left E-module. Since E contains the left multiplications $\boldsymbol{\gamma }_a$, every E-submodule of L is a left ideal of the field L; the E-module L is therefore simple. Every endomorphism of the additive group of L that commutes with the $\boldsymbol{\gamma }_a$ is of the form $\boldsymbol{\delta }_b:x\mapsto xb$ with $b$ in L. Consequently, $b\mapsto \boldsymbol{\delta }_b$ is an isomorphism from K to the opposite ring of End$_E$(L), which is a field.

The bicommutant $E''$ of the E-module L therefore consists of the endomorphisms of the right K-vector space L. Let $v$ be an endomorphism of the K-vector space L whose restriction to V coincides with $h$; it is an element of $E''$. Let $(x_i)_{i\in I}$ be a basis of V over K; by Theorem 3 (VIII, p. 88), there exists an element $u$ of E such that $u(x_i) =v(x_i) =h(x_i)$ for $i\in I$. By linearity, it follows that $u(x) =h(x)$ for every $x$ in V.