Vector Spaces

Abstract

As usual, a collection of objects will be called a set. A member of the collection is also called an element of the set. It is useful in practice to use short symbols to denote certain sets. For instance we denote by R the set of all numbers. To say that “x is a number” or that “x is an element of R ⁿ amounts to the same thing. The set of n-tuples of numbers will be denoted by W. Thus “X is an element of R ⁿ and ”X is an n-tuple“ mean the same thing. Instead of saying that u is an element of a set S, we shall also frequently say that u lies in S and we write u E S. If S and S’ are two sets, and if every element of S’ is an element of S, then we say that S’ is a subset of S. Thus the set of rational numbers is a subset of the set of (real) numbers. To say that S is a subset of S’ is to say that S is part of S’. To denote the fact that S is a subset of S’, we write S S’.