Three of the basic properties of vectors in 3 dimensions can be generalized to functions: (i) addition and subtraction, (ii) scalar multiplication, and (iii) the dot product. The dot product for two functions is the integral of the appropriate product of the functions.

Just as the x, y, z unit vectors form an orthonormal basis in 3 dimensions, one can also find functions which form an orthonormal basis in the function vector space. Here one needs an infinite number of functions. Orthonormal means that the dot product of two basis different vectors is zero, and the dot product of a basis vector with itself is one. The condition that any vector can be written as linear combinations of basis vectors is called completeness. We give explicitly one complete orthonormal basis in function space, a set of sine functions defined on the interval [0,L].