The book contains a theory that allows you to evaluate various sections of numerical methods from a single point of view. Optimal approximations are considered and effective methods for their production are given. The theory is illustrated by numerous examples from the field of interpolation, numerical integration, etc. Of particular note is the Fourier synthesis and the recently attracted theory of spline functions. The theory is based on functional analysis. All the necessary information from the functional analysis is given in the text. The book is intended for all interested in theories and practice of numerical methods.