Basics Of Functional Analysis With Bicomplex — Sc...

[ \mathbbBC = z_1 + z_2 \mathbfj \mid z_1, z_2 \in \mathbbC ]

[ \mathbbBC = (z_1, z_2) \mid z_1, z_2 \in \mathbbC ] Basics of Functional Analysis with Bicomplex Sc...

A is defined as: [ |w|_\mathbfk = \sqrtw \cdot \barw = \sqrt(z_1 + z_2 \mathbfj)(\barz_1 - z_2 \mathbfj) = \sqrt z_1 \barz_1 + z_2 \barz_2 + \mathbfk (z_2 \barz_1 - z_1 \barz_2) ] which takes values in ( \mathbbR \oplus \mathbbR \mathbfk ) (the hyperbolic numbers). But careful: this is not real-valued. To get a real norm, one composes with a “hyperbolic absolute value.” [ \mathbbBC = z_1 + z_2 \mathbfj \mid

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 ). Every bicomplex number has a unique : This

Every bicomplex number has a unique :

This decomposition is the key to bicomplex analysis: it reduces bicomplex problems to two independent complex problems . In classical functional analysis, we work with vector spaces over ( \mathbbR ) or ( \mathbbC ). Over ( \mathbbBC ), a bicomplex module replaces the vector space, but caution: ( \mathbbBC ) is not a division algebra (it has zero divisors, e.g., ( \mathbfe_1 \cdot \mathbfe_2 = 0 ) but neither factor is zero). Hence, we cannot define a bicomplex-valued norm in the usual sense—the triangle inequality fails due to zero divisors.