Orthogonal Functions - Generalization of Vectors

Generalization of Vectors

It can be shown that orthogonality of functions is a generalization of the concept of orthogonality of vectors. Suppose we define V to be the set of variables on which the functions f and g operate. (In the example above, V={x} since x is the only parameter to f and g. Since there is one parameter, one integral sign is required to determine orthogonality. If V contained two variables, it would be necessary to integrate twice--over a range of each variable--to establish orthogonality.) If V is an empty set, then f and g are just constant vectors, and there are no variables over which to integrate. Thus, the equation reduces to a simple inner-product of the two vectors.