Let $L$ be a line bundle on a paracompact strictly $K$-analytic space $V$ and let $F/K$ be a non-archimedean field extension.

• (a)

  The base change of a piecewise linear (resp. piecewise ${\mathbb{Q}}$-linear) metric on $L$ is a piecewise linear (resp. piecewise ${\mathbb{Q}}$-linear) metric on $L\hat{\otimes}_{K}F$.

• (b)

  If $F$ is a subfield of ${\mathbb{C}}_{K}$ and if $V$ is compact, then every piecewise linear (resp. piecewise ${\mathbb{Q}}$-linear) metric on $L\hat{\otimes}_{K}F$ is the base change of a unique piecewise linear (resp. piecewise ${\mathbb{Q}}$-linear) metric on $L\hat{\otimes}_{K}{K^{\prime}}$ for a suitable finite subextension $K^{\prime}/K$ of $F/K$.