Basics Of Functional Analysis With Bicomplex Sc... Instant

[ | \lambda x | = |\lambda| \mathbbC | x | \quad \textor more generally \quad | \lambda x | = |\lambda| \mathbbBC | x | ? ] But ( |\lambda|_\mathbbBC = \sqrtz_1 ) works, giving a real norm. However, to preserve the bicomplex structure, one uses :

[ w = z_1 + z_2 \mathbfj = \alpha \cdot \mathbfe_1 + \beta \cdot \mathbfe_2 ] where [ \mathbfe_1 = \frac1 + \mathbfk2, \quad \mathbfe_2 = \frac1 - \mathbfk2 ] satisfy ( \mathbfe_1^2 = \mathbfe_1, \ \mathbfe_2^2 = \mathbfe_2, \ \mathbfe_1 \mathbfe_2 = 0, \ \mathbfe_1 + \mathbfe_2 = 1 ), and ( \alpha = z_1 - i z_2, \ \beta = z_1 + i z_2 ) are complex numbers. Basics of Functional Analysis with Bicomplex Sc...

Any bicomplex Banach space ( X ) is isomorphic (as a real Banach space) to ( X_1 \oplus X_2 ), where ( X_1, X_2 ) are complex Banach spaces, and bicomplex scalars act by: [ (z_1 + z_2 \mathbfj) (x_1 \mathbfe_1 + x_2 \mathbfe_2) = (z_1 - i z_2) x_1 \mathbfe_1 + (z_1 + i z_2) x_2 \mathbfe_2. ] [ | \lambda x | = |\lambda| \mathbbC