ABSTRACT
The upshot of all of this is that we can call a basis of L2[0, 1] in that any
element of L2[0, 1] is a linear combination of . This is the basis of a Taylor series or Fourier series representation of a function. Now recall that the basis of an n-dimensional vector space is a set of n orthogonal vectors {ui} and that any vector in that space can be written as a linear sum of those vectors: i.e., any vector x is
where the a i’s are scalars. Recall also that
Therefore, we are tempted to identify the basis functions of L2[0, 1] with the basis of a vector space. However, there is one important difference: while the vector spaces that we are familiar with all have finite dimensions, the number of basis functions of L2[0, 1] is infinite.
