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

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 ${\mathbb{C}}_2[\xi {]}_n$. 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.