Sesquilinear A-form and Bilinear A-form

In this paper, we introduce and investigate the concepts of sesquilinear Aforms and bilinear A-forms, which extend the classical notions of sesquilinear and bilinear forms to modules over C * -algebras.Rigorous definitions for both left and right sesquilinear A-forms, as well as bilinear A-forms, are provided together with illustrative examples.Several structural properties of these A-forms are established, including their relationships with corresponding quadratic A-forms and inequalities analogous to the Cauchy-Schwarz inequality.The results presented here form a foundational framework for further research in A-valued functional analysis and operator theory.