Module 2 : Fundamentals of Vector Spaces
Section 2 : Vector Spaces
 
2. Vector Spaces

The concept of a vector space will now be formally introduced. This requires the concept of closure and field.

Definition 1 (Closure)
A set is said to be closed under an operation when any two elements of the set subject to the operation yields a third element belonging to the same set.

Example 2
The set of integers is closed under addition, multiplication and subtraction. However, this set is not closed under division.

Example 3
The set of real numbers ($R$) and the set of complex numbers (C) are closed under addition, subtraction, multiplication and division.

Definition 4 (Field)
A field is a set of elements closed under addition, subtraction, multiplication and division

Example 5
The set of real numbers ($R$) and the set of complex numbers ($C$) are scalar fields. However, the set of integers is not a field.

A vector space is a set of elements, which is closed under addition and scalar multiplication. Thus, associated with every vector space is a set of scalars $F$ (also called as scalar field or coefficient field) used to define scalar multiplication on the space. In functional analysis, the scalars will be always taken to be the set of real numbers ($R$) or complex numbers ($C$).