This paper is the natural sequel to [1]. An inner product is defined for the linear vector space [S] of all the piecewise linear (PWL) continuous mappings defined over a rectangular compact set S using a simplicial partition This permits us to assure that [S] is a Hilbert space.Then, the notion of orthogonality is introduced and orthonormal bases of PWL functions are formulated. A relevant consequence of the approach is that the problem of function approximation can be translated to the more studied field of approximation in Hilbert spaces of finite dimension. As will be shown, this powerful theoretical framework can be used to generate an optimal scheme for model reduction.

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