ABSTRACT

The existence of norms is not the unique property of the Euclidean spaces. In particular, there is no immediate way one can talk about angles between vectors on a general normed space, a central geometric feature of the Euclidean structure. This chapter is devoted to the study of infinite-dimensional normed spaces which admit more structure than a mere norm and on them we may study geometric questions, like angles between vectors. The chapter concerns ourselves with the study of inner product spaces. These are vector spaces endowed with an inner product, which is a function assigning a certain scalar to any pair of vectors. The definition of an inner product on a vector space, which, as pointed out in the introduction, represents an axiomatisation of some essential properties of the familiar dot product in Euclidean spaces. Throughout the chapter, any inner product space will automatically be considered as a normed space with the induced norm.