New Proofs on Properties of an Orthogonal Decomposition of a Hilbert Space

TL;DR

This paper presents novel proofs of properties related to the orthogonal decomposition of Hilbert spaces, focusing on projections, angles between functions, and relationships within Sobolev spaces.

Contribution

It introduces new proof techniques for properties of Hilbert space decompositions, including geometric interpretations and relationships between function spaces.

Findings

New proofs of projection properties in Hilbert spaces

Analysis of angles between functions and subspaces

Insights into relationships within Sobolev spaces

Abstract

We establish new and different kinds of proofs of properties that arise due to the orthogonal decomposition of the Hilbert space, including projections, over the unit interval of one dimension. We also see angles between functions, particularly between those which are non zero constant multiples of each other and between functions from the kernel space and the derivative image of the trace less Sobolev space.