On Orthogonal Decomposition of a Hilbert Space

TL;DR

This paper presents an orthogonal decomposition of a Hilbert space into null solutions of derivatives within higher order Hilbert/Sobolev spaces, including properties of projections and applications to elementary functions.

Contribution

It introduces a novel orthogonal decomposition framework for Hilbert spaces involving derivatives in higher order Sobolev spaces.

Findings

Decomposition of Hilbert space into null solutions of derivatives

Properties of orthogonal projections in this context

Examples of decomposing elementary functions

Abstract

In this short article we show an orthogonal decomposition of a Hilbert space as a sum of null solutions of the first derivative and the first derivative of a traceless higher order Hilbert/Sobolev space. We define orthogonal projections and see some of its properties and show case some decomposition of elementary functions as corollaries.