Basics Of Functional Analysis With Bicomplex Sc... [work] Jun 2026

A complete bicomplex normed space (with respect to the topology induced by the Euclidean norm (\sqrta^2+b^2) from the idempotent components) is called a .

The core of functional analysis lies in the concept of "size" or distance. In standard analysis, a norm maps a vector to a non-negative real number. In bicomplex functional analysis, we require a norm that interacts correctly with the bicomplex scalar field. Basics of Functional Analysis with Bicomplex Sc...

A bicomplex module $X$ is a set of elements (vectors) where addition and scalar multiplication by bicomplex numbers are defined. The linearity properties (associativity, distributivity) remain intact due to the commutativity of $\mathbbBC$. A complete bicomplex normed space (with respect to

For over a century, Functional Analysis has been built upon the sturdy foundation of real and complex numbers. The transition from real to complex scalars was not merely a curiosity; it unlocked the spectral theorem for operators, simplified matrix algebra, and provided the natural setting for quantum mechanics. The completeness of the complex field (\mathbbC) is a cornerstone of Banach and Hilbert space theory. In bicomplex functional analysis, we require a norm

(with respect to (j)): (z = w_1 + w_2 i), where (w_1, w_2 \in \mathbbC_j).

This article explores the foundational concepts of this emerging field. We will cover the algebra of bicomplex numbers, the definition of bicomplex normed spaces, linear operators, and the crucial notion of bicomplex Hilbert spaces. Along the way, we’ll see why this generalization is non-trivial and where it might lead.