Basics Of Functional Analysis With Bicomplex Sc... Jun 2026

The clean approach: Use the idempotent decomposition. For ( x \in X ) (bicomplex module), write ( x = x_1 \mathbfe_1 + x_2 \mathbfe_2 ) with ( x_1, x_2 ) in a complex Banach space ( E ). Then define a real norm as: [ | x | = \max( | x_1 |, | x_2 | ) ] or ( | x | = \sqrt^2 ). This makes ( X ) a real Banach space but retains bicomplex scalar multiplication via the idempotents.

Further Reading Suggestions:

As we move beyond classical complex analysis, the bicomplex setting invites us to rethink fundamentals: What does it mean for a space to be "complete"? How do zero divisors affect the notion of eigenvalue? And might this lead to a unified treatment of two-component physical systems in quantum mechanics? Basics of Functional Analysis with Bicomplex Sc...

, as scalars. The framework establishes a theory of bicomplex modules, inner products, and linear operators, extending fundamental results like the Hahn-Banach theorem and Riesz representation to this setting. For a comprehensive study, read more at The clean approach: Use the idempotent decomposition