Let $V$ be a paracompact strictly $K$-analytic space with a line bundle $L$. Then the following properties hold:

• (a)

  A piecewise ${\mathbb{Q}}$-linear metric on $L$ is continuous.

• (b)

  The isometry classes of piecewise linear (resp. piecewise ${\mathbb{Q}}$-linear) metrics on line bundles of $V$ form an abelian group with respect to $\otimes$.

• (c)

  The pull-back $f^{*}{\|\hskip 4.30554pt\|}$ of a piecewise linear (resp. piecewise ${\mathbb{Q}}$-linear) metric ${\|\hskip 4.30554pt\|}$ on $L$ with respect to a morphism $f:W\to V$ of paracompact analytic spaces is a piecewise linear (resp. piecewise ${\mathbb{Q}}$-linear) metric on $f^{*}L$.

• (d)

  The minimum and the maximum of two piecewise linear (resp. piecewise ${\mathbb{Q}}$-linear) metrics on $L$ are again piecewise linear (resp. piecewise ${\mathbb{Q}}$-linear) metrics on $L$.