International Journal For Multidisciplinary Research

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Volume 8 Issue 4
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Bicomplex-Valued Multi-Norm Spaces: Foundations and Structural Analysis

Author(s) Neetu Singh
Country India
Abstract The term "bicomplex-valued norm" necessitates qualification due to the absence of a compatible total order in the bicomplex algebra. The positive hyperbolic cone, which is derived from the idempotent modulus, more precisely values a norm that is appropriate for bicomplex modules. This paper provides a structural and foundational theory for multi-norm spaces. A bicomplex module X=e₁X₁⊕e₂X₂ is equipped with Nₙ=Nₙ,1e₁+Nₙ,2e₂ at each level, where the components are classical complex multi-norms. The componentwise proof of the multi-norm axioms, positivity, definiteness, triangle inequality, and scalar domination is presented, as well as a converse decomposability characterisation. The product topology is demonstrated to be equivalent to the hyperbolic-ball topology; convergence, Cauchy sequences, completeness, and fixed-level equivalence are precisely equivalent to the two complex components. Closed subspaces, product modules, minimum quotient multi-norms, continuous duals, and finite-dimensional modules yield structural results. The analysis elucidates the rationale behind the legitimate existence of nonzero null-cone norms in pure idempotent vectors and the exclusion of natural submodules by demand for invertible norm values. The failure of bicomplex homogeneity by a real Euclidean scalarization and the inability of an unordered bicomplex codomain to support norm inequalities are demonstrated by counterexamples. Bicomplex Banach multi-modules are generated as paired complex Banach multi-normed spaces. Applications and open questions pertain to C*-type modules, weak topologies, tensor products, quotient constructions beyond the minimum multi-norm, and duality.
Keywords bicomplex multi-norm; hyperbolic modulus; positive cone; product topology; quotient space; continuous dual; Banach multimodule
Published In Volume 8, Issue 4, July-August 2026
Published On 2026-07-29
DOI https://doi.org/10.36948/ijfmr.2026.v08i04.85024
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