On The Adjoint of Bounded Operators On A Semi-Inner Product Space

Abstract

The notion of semi-inner product (SIP) spaces is a generalization of inner product (IP) spaces notion by reducing the positive definite property of the product to positive semi-definite. As in IP spaces, the existence of an adjoint of a linear operator on a SIP space is guaranteed when the operator is bounded. However, in contrast, a bounded linear operator on SIP space can have more than one adjoint linear operators. In this article we give an alternative proof of those results using the generalized Riesz Representation Theorem in SIP space. Further, the description of all adjoint operators of a bounded linear operator in SIP space is identified.