Solution: Define a as a map ( | \cdot | : X \to \mathbbR_+ ) satisfying standard Banach space axioms, but with scalar multiplication by bicomplex numbers respecting:
[ | \lambda x | = |\lambda| \mathbbC | x | \quad \textor more generally \quad | \lambda x | = |\lambda| \mathbbBC | x | ? ] But ( |\lambda|_\mathbbBC = \sqrt^2 ) works, giving a real norm. However, to preserve the bicomplex structure, one uses : Basics of Functional Analysis with Bicomplex Sc...
But here’s the crucial difference from quaternions: ( i \mathbfj = \mathbfj i ) (commutative). Then ( (i \mathbfj)^2 = +1 ). Define the hyperbolic unit ( \mathbfk = i \mathbfj ), so ( \mathbfk^2 = 1 ), ( \mathbfk \neq \pm 1 ). Solution: Define a as a map ( |
It sounds like you’re looking for a feature article or an in-depth explanatory piece on (likely short for Bicomplex Scalars or Bicomplex Numbers ). Then ( (i \mathbfj)^2 = +1 )