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.