Inner Products and Banach Algebra structures on Bicomplex Numbers and Their Associated Spaces

Abstract

In this paper, we introduce various types of inner products and norms on the bicomplex number system $\mathbb{C}_2$, the bicomplex vector space ${\mathbb{C}_2}^{n}$, the space of bicomplex matrices ${C_2}^{m \times n}$, and the space of bicomplex polynomials $\mathbb{C}_2[ξ]_n$. We investigate the relationships among these inner products and norms, and establish several results. Furthermore, we prove that $\mathbb{C}_2$ and ${\mathbb{C}_2}^{n}$ are Banach algebras and Hilbert spaces. These results provide a unified framework for the study of inner product structures and normed linear spaces over bicomplex numbers and their associated spaces.