CHARACTERIZATION OF CAUCHY-SCHWARZ INEQUALITY WITH TAPIA SEMI-INNER PRODUCT ON COMPLEX INNER PRODUCT SPACES

Abstract

Tapia semi-inner product is defined on a normed space. If a normed space is constructed by an inner product, then the Tapia semi-inner product is equivalent to that inner product. This research constructs a refinement of the Cauchy-Schwarz inequality using the Tapia semi-inner product. Furthermore, the characterization of the Cauchy-Schwarz inequality is extended to complex inner product spaces with specific adjustments. The results provide a new perspective on triangle inequality characterization in complex domains.