Basis function

In functional analysis and its applications, a function space can be viewed as a vector space of infinite dimension whose basis vectors are the basis functions. This means that each function of the function space can be composed by a linear combination of the basis functions.

In an easier way, it can also be expressed like this:

Let's begin explaining from a simple example : you can create any (two-dimensional) vector by adding vectors (1,0) and (0,1), each multipied by a given number:

In this example, vectors (1,0) and (0,1) are basis vectors for the vector (x,y). The best basis vectors are perpendicular or orthogonal to each other, which is fulfilled by (1,0) and (0,1).

In functions, we have function f(x) instead of vector (x,y). Imagine a musical tone - it can be represented by a sum of sines and cosines with various amplitudes and frequencies. In this example, sines and cosines are basis functions (used in Fourier transform).