On an energy equality in the theory of evolution equations

TL;DR

This paper provides a straightforward proof of a fundamental energy equality in the theory of evolution equations, clarifying the foundational mathematical relationships in this area.

Contribution

It offers an elementary proof of an essential energy equality, enhancing understanding of evolution equations' theoretical framework.

Findings

Simplifies the proof of the energy equality in evolution equations

Clarifies the mathematical setting involving V, H, and their duals

Strengthens theoretical foundations of evolution equations

Abstract

We give an elementary proof of a basic "energy" equality in the theory of evolution equations in the setting, which usually starts with introducing $V\subset H=H^{*}\subset V^{*}$.