Bicomplex Sequence Spaces Generated by Multi-Norms
Author(s): Neetu Singh
Publication #: 2608026
Date of Publication: 25.08.2026
Country: India
Pages: 1-9
Published In: Volume 12 Issue 4 August-2026
DOI: https://doi.org/10.62970/IJIRCT.v12.i4.2608026
Abstract
Classical coordinate methods, idempotent-valued norms, and zero-divisor algebra are combined in bicomplex sequence spaces. The spaces c₀(ℬℂ), c(ℬℂ), ℓᵖ(ℬℂ), and ℓ∞(ℬℂ) are introduced in this paper. Subsequently, sequence spaces are generated by a bicomplex multi-norm. A pair of complex sequence spaces is isometrically identified with each classical bicomplex space. The following are derived componentwise: completeness, inclusion relations, density of finitely supported sequences, coordinate projections, shifts, Schauder bases, continuous duals, and Köthe duals. The inequality ‖x‖q,????≼????‖x‖p,???? is established for counting-measure sequence spaces when 1≤p≤q≤∞. The domain dependence of the inequality is underscored. Using sup_n Nₙ(x₁,…,xₙ), a prefix-bounded multi-norm space m_N(E) is defined and proven to be Banach whenever E is complete. The minimum multi-norm is demonstrated to reduce to c₀(E) in the analysis of a tail-vanishing subspace. Operator-generated weighted spaces, matrix transformations, and a coordinate-tail criterion for compact operators are established. The unit-vector sequence in ℓ¹ serves as a counterexample, while weak convergence is characterised for reflexive ℓᵖ ranges and c₀. Geometric, basis, shift, diagonal, matrix, and multi-norm-generated sequences are among the examples that have been implemented. Open problems pertain to the spectral properties of sequence operators, weak compactness, bicomplex matrix domains, Köthe echelon constructions, and nonminimum multi-norm duality.
Keywords: bicomplex sequence space; ℓp space; multi-norm-generated sequence; Schauder basis; Köthe dual; matrix transformation; compact operator; weak convergence
Download/View Count: 3
Share this Article