As mentioned in the previous article, a vector space is a set of objects that follows certain rules of addition and multiplication. This implies that a set of functions that follows the same rules, forms a ** vector space of functions**. The properties of a vector space of functions are:

1) Commutative and associative addition for all functions of the closed set .

2) Associativity and distributivity of scalar multiplication for all functions of the closed set.

where and are scalars.

3) Scalar multiplication identity.

4) Additive inverse.

5) Existence of null vector , such that

where in this case is a zero function that returns zero for any inputs.

Similarly, a set of linearly independent functions forms a set of ** basis functions**. We have a complete set of basis functions if any well-behaved function in the domain can be written as a linear combination of these basis functions, i.e.

In quantum chemistry, physical states of a system are expressed as functions called ** wavefunctions**.