Natural Isomorphism from a Linear Map's Image to Complement of Nullspace

TL;DR

This paper establishes a natural isomorphism between a linear map's image and the complement of its nullspace in the context of Banach and Hilbert spaces, extending Riesz representation and aiding fluid dynamics equations.

Contribution

It introduces a generalization of the Riesz representation theorem to bounded linear maps from Banach to Hilbert spaces, providing a new tool for solving pressure equations in fluid dynamics.

Findings

Established a natural isomorphism between image and complement of nullspace.

Generalized Riesz representation theorem for bounded linear maps.

Applied the isomorphism to solve pressure equations in fluid dynamics.

Abstract

There is a natural isomorphism from image to complement of nullspace, for a bounded linear map from a real Banach space onto a closed subspace of a real Hilbert space. This generalizes Riesz representation (self-duality of Hilbert space). The isomorphism helps solve the pressure equation of fluid dynamics.