Some Fundamental Theorems of Functional Analysis with Bicomplex and Hyperbolic Scalars

Abstract

We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem and the Hahn Banach separation theorem are proved.

Integration of Bicomplex Valued Function along Hyperbolic Curve

“…Then we have two complex planes C (i 1 ) = {x + i 1 y : x, y ∈ R} and C (i 2 ) = {x + i 2 y : x, y ∈ R} , both of which are identical to C. Bicomplex numbers( [1], [13]) are defined as…”

Hyperbolic Functions of Bounded Variation and Riemann-Stieltjes Integral involving Strong Partitions of Hyperbolic Intervals

“…Even when there are elements into the set that are no related, all of them are related with the supremum by the partial order. For a fuller treatment we refer the reader to [7,8,17,18].…”

Hyperbolic Valued Metric Space

“…Corrado Segre [7] introduced bicomplex numbers in 1982. The set of bicomplex numbers [1], [6] is defined as…”