We begin with the definition of a vector space. Where appropriate, we will give simpler definitions, which at the expense of some generality will be sufficient for the use made of them in the text. For example a vector space can be defined over any field, but we will consider vector spaces over the real numbers, so that what we now introduce are sometimes called ‘real vector spaces’.
Definition B.1 A set X is a vector space (VS) if two operations (addition, and multiplication by scalar) are defined on X such that, for x, y X, and ,
and such that in addition X is a commutative group with identity ...