On the approximate minimization of functionals

This paper considers in general the problem of finding the minimum of a given functional f (u) f (u) over a set B B by approximately minimizing a sequence of functionals f n (u n) fₙ (uₙ) over a "discretized" set B n Bₙ ; theorems are given proving the convergence of the approximating points u n uₙ in B n Bₙ to the desired point u u in B B. Applications are given to the Rayleigh-Ritz method, regularization, Chebyshev solution of differential equations, and the calculus of variations.