Research on Koopman operator theory has focused on three key areas for several decades: the mathematical structure of the Koopman eigenfunction space, the basis of this space, and the ability to represent nonlinear dynamics as linear. This study provides a thorough and comprehensive framework for these topics, including theoretical, analytical, and numerical approaches. A novel mathematical structure is introduced, which outlines permissible actions on the infinite set of Koopman Eigenfunction, under which this set is closed. Notions of generating and independent sets of Koopman eigenfunctions are defined. In addition, notions of a minimal generating set, and a maximal independent set are defined and are shown to be equivalent. This structure defines conditions for independence within the set of Koopman eigenfunctions. This independent set can be interpreted as a new coordinate system in which the dynamical system is linear. The theory also highlights the equivalence of a minimal set, flowbox representation, and conservation laws. Finally, the presented theory is supported by numerical experiments.

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