Orthogonality and Bases

Abstract

Let X be a linear space over K and suppose that on X we have an inner product 〈,〉. The basic notion defined on X by the inner product 〈,〉 is the notion of orthogonality. Using this notion we consider certain families of orthogonal elements which are on the surface of the unit ball of X (considered as a normed linear space (X, ||, ||). Further, we present the notion of c-completeness and the notion of base in complete inner product spaces. Also certain results about subspaces in X and the orthogonal decomposition of a complete inner product space are given and the dual space is then examined. We also present some examples of nonseparable complete inner product spaces.