Approximation problem in Hilbert space

In this section we will illustrate how one can use the geometrical properties of the Hilbert space to solve an approximation problem. Suppose that L is an n dimensional subspace of a Hilbert space H L C H) and L is spanned by a linearly independent 

To solve this problem let us consider the norm of difference Udo—d l. Any vector d e L can be represented in the form of a linear combination of basis vectors  

Let us calculate the derivatives of the do—d with respect to which must vanish at an extremum point  

From the last equation we have the system of linear equations for the unknown coefficients a  

Functional spaces of geophysical models euid data 

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