Extension of isometries in real Hilbert spaces

TL;DR

This paper develops a theory for extending local isometries in real Hilbert spaces, showing that such extensions are possible without convexity or openness constraints, thus advancing the understanding of isometry extensions.

Contribution

It introduces generalized linear spans and proves the extension of local isometries to entire Hilbert spaces without convexity or openness restrictions.

Findings

Local isometries can be extended to entire Hilbert spaces.

Extensions do not require the domain to be convex or open.

The theory improves upon previous results by removing domain restrictions.

Abstract

In this paper, the notions of first-order and second-order generalized linear spans and index set are defined. Moreover, their properties are investigated and applied to the studies of extension of isometries. We develop the theory of extending the domain of local isometries to the generalized linear spans, where we call an isometry defined in a subset of a Hilbert space a local isometry. In addition, we prove that the domain of local isometry can be extended to any real Hilbert space, where the domain of local isometry does not have to be a convex body or an open set. This indicates that the main results of this paper are superior to those previously published.