TWO - INTRODUCTION TO FUNCTIONAL ANALYSIS. DISTRIBUTIONS

Abstract

There are different ways in which a sequence of real- or complex-valued functions on an interval can converge to a limit function. Pointwise convergence, that is, convergence of the sequence of numbers to the number at every point, is one of the least important ways. In the case of most of the more important kinds of convergence, such as uniform convergence and convergence in the mean of orders 1 and 2, there is a way of measuring how rapidly a sequence of functions converges to a limit function. This is done with the aid of a distance concept on a set in which the functions are considered as points or vectors. Such a distance concept makes it possible to discuss approximation problems in a quantitative manner. In a space with a distance function—a metric space—one has a certain geometry. This geometry is sometimes so strange that it is of little help in solving problems. In normed vector spaces of finite dimension, the geometry is fairly reasonable; strange phenomena occur in the case of infinite dimensional spaces. The geometry is still better in scalar product spaces. In these spaces length and distance are defined in terms of the scalar product. Approximation in the mean of order 2 is convenient because it can be interpreted as approximation in a scalar product space.