The structure of inner product

Abstract

Inner product spaces are special normed spaces that allow norm (or length) as well as angle to be expressed in terms of the inner product. Every inner product gives rise to a norm. However, not every norm is generated by some inner product. In this project, we investigate the structures of inner products on finite-dimensional spaces as well as general normed spaces. For finite dimensional spaces, we shall mainly search for the relations between different inner products, specifically how to construct inner products from a given one. For normed spaces, we shall discuss various conditions for a given normed space to be an inner product space.