On the notion of scalar product for finite-dimensional diffeological vector spaces

TL;DR

This paper introduces the concept of pseudo-metrics for finite-dimensional diffeological vector spaces to address the limitations of smooth scalar products, and explores their properties and implications for subspaces and dual spaces.

Contribution

It proposes pseudo-metrics as a way to generalize scalar products in diffeological vector spaces and analyzes their effects on subspace splitting and dual space structures.

Findings

Pseudo-metrics are the least-degenerate symmetric bilinear forms on diffeological spaces.

Not all subspaces of a diffeological vector space split smoothly as direct summands.

The diffeological dual always has the standard diffeology, and pseudo-metrics induce smooth scalar products on duals.

Abstract

It is known that the only finite-dimensional diffeological vector space that admits a diffeologically smooth scalar product is the standard space of appropriate dimension. In this note we consider a way to circumnavigate this issue, by introducing a notion of pseudo-metric, which, said informally, is the least-degenerate symmetric bilinear form on a given space. We apply this notion to make some observation on subspaces which split off as smooth direct summands (providing examples which illustrate that not all subspaces do), and then to show that the diffeological dual of a finite-dimensional diffeological vector space always has the standard diffeology and in particular, any pseudo-metric on the initial space induces, in the obvious way, a smooth scalar product on the dual.