A pointwise norm on a non-reduced analytic space

Journal article, 2022

Let X be a possibly non-reduced space of pure dimension. We introduce a pointwise Hermitian norm on smooth (0,q)-forms, in particular on holomorphic functions, on X. The norm is is canonical, up to equivalence, where the underlying reduced space is a manifold. We prove that the space of holomorphic functions is complete with respect to the natural topology induced by this norm.