When the values of a function f are too difficult to calculate, whether "by hand" or with a tool such as a calculator or computer, we try to replace f with a simpler function—a polynomial, for example. Of course, this replacement must be "close" to f. We therefore need to specify what we mean by "close"—that is, choose a distance.
The most natural approach
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Let f and g be functions that are defined and, for simplicity, assumed continuous on the interval [a, b]. The best-known distance between f and g is probably the one induced by the uniform norm: d(f, g) is the maximum difference between f(x) and g(x) as x ranges over [a, b]. Mathematically, this is written:
We must, of course, show that this definition has the properties of a distance (see Les distances. Bibliothèque Tangente no. 81, POLE, 2023). No problem: it does! To approximate f with a simpler function g, we therefore seek to make d(f, g) as small as possible. This is clearly the appropriate distance if we want a function g that is close to f at every point of the interval [a, b*].
