Orthonormal Bases and Fourier Representations in Euclidean Space

We discuss orthonormal bases of Euclidean Space. We prove that an orthonormal collection of vectors is always linearly independent and therefore a sufficient number of them will form a basis of a finite dimensional vector space. An immediate consequence of the linear independence argument is the the fact that the coefficients in the orthonormal basis are easy to compute; they are the Fourier coefficients.