International Journal For Multidisciplinary Research

E-ISSN: 2582-2160   •   Impact Factor: 9.24

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A Comprehensive Study of Hyperbolic-Valued Multi-Norms: Definitions, Properties, and Characterizations

Author(s) Neetu Singh
Country India
Abstract An ordinary real-valued norm cannot be transferred to a hyperbolic module without specifying an ordered codomain and a decomposition formula, as hyperbolic numbers form a commutative real algebra with nontrivial idempotents and zero divisors. This paper establishes a rigorous framework for multi-norms that take values in the positive hyperbolic cone. The construction commences with the idempotent representation of a hyperbolic module and two classical multi-normed component spaces. Afterward, a hyperbolic-valued multi-norm is componentwise defined and demonstrated to satisfy the triangle inequality, scalar domination, zero stability, repetition consistency, and permutation invariance. The decomposability conditions under which every hyperbolic multi-norm arises from a pair of real multi-norms are identified by a converse characterisation. Componentwise criteria are established for completeness, equivalence, convergence, Cauchy behaviour, and comparison. Concrete examples of finite-dimensional modules and function spaces are computed, and sum and Euclidean tuple gauges are employed as counterexamples to illustrate that norm-like aggregation alone does not imply the repetition axiom. The analysis demonstrates that idempotent coordinates should be employed to address partial order and zero-divisor effects, rather than unrestricted hyperbolic division. The discussion includes applications to product systems, stability estimates, and split-channel models, without the assumption of empirical validation. Open questions pertain to compactness, operator-generated constructions, uniform equivalence across levels, tensor products, and dual multi-norms.
Keywords hyperbolic numbers; multi-normed space; idempotent decomposition; positive hyperbolic cone; zero divisors; completeness; functional analysis
Published In Volume 6, Issue 4, July-August 2024
Published On 2024-08-07
DOI https://doi.org/10.36948/ijfmr.2024.v06i04.85578
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CrossRef DOI prefix of IJFMR is 10.36948/ijfmr

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