Bicomplex Sc...: Basics Of Functional Analysis With

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 ).

This decomposition is the of the theory: every bicomplex functional analytic result follows from applying complex functional analysis to each idempotent component. 4. Bicomplex Linear Operators Let ( X, Y ) be bicomplex Banach spaces. A map ( T: X \to Y ) is bicomplex linear if: [ T(\lambda x + \mu y) = \lambda T(x) + \mu T(y), \quad \forall \lambda, \mu \in \mathbbBC, \ x,y \in X. ] Basics of Functional Analysis with Bicomplex Sc...

with componentwise addition and multiplication. Equivalently, introduce an independent imaginary unit ( \mathbfj ) (where ( \mathbfj^2 = -1 ), commuting with ( i )), and write: It sounds like you’re looking for a feature

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 ). Bicomplex Linear Operators Let ( X, Y )