Differentiable mappings between weighted restricted products

TL;DR

This paper develops a framework for analyzing the continuity and smoothness of mappings between weighted restricted products of locally convex spaces, with applications to infinite-dimensional Lie groups of diffeomorphisms.

Contribution

It introduces restricted products for locally convex spaces and establishes criteria for the continuity and differentiability of mappings into these products, especially for weighted function spaces.

Findings

Criteria for continuity of mappings into restricted products

Conditions for differentiability of non-linear mappings between weighted function spaces

Foundation for studying weighted vector fields on Riemannian manifolds

Abstract

In this paper, we introduce restricted products for families of locally convex spaces and formulate criteria ensuring that mappings into such products are continuous or smooth. As a special case, can define restricted products of weighted function spaces and obtain results concerning continuity and differentiability properties of natural non-linear mappings between such spaces. These concepts and results are the basis for the study of weighted vector fields on Riemannian manifolds in a subsequent work (see [B. Walter, "Weighted diffeomorphism groups of Riemannian manifolds", arXiv: 1601.02834]), which serve as modelling spaces for suitable infinite-dimensional Lie groups of diffeomorphisms.