On Orthogonal Decomposition of a Sobolev Space

TL;DR

This paper investigates the orthogonal decomposition of Sobolev spaces, analyzing properties of the inner product, the associated distance, and how space expansions relate to increasing regularity.

Contribution

It introduces a detailed analysis of the orthogonal decomposition in Sobolev spaces, including the dimension of difference spaces and space expansion properties with regularity.

Findings

Determined the dimension of the orthogonal difference space.

Analyzed properties of the inner product and the induced distance.

Showed how Sobolev spaces expand as regularity increases.

Abstract

The theme of this short article is to investigate an orthogonal decomposition of a Sobolev space and look at some properties of the inner product therein and the distance defined from the inner product. We also determine the dimension of the orthogonal difference space and show the expansion of spaces as their regularity increases.