An orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces using the GramSchmidt process (a method for orthonormalizing a set of vectors in an inner product space), in every finite-dimensional inner product space.
In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the GramSchmidt process. 
