The Uniform Metric

We wish to define the notion of distance between any two functions (continuous for simplicity) on an interval I.

The norm of a function on an interval

We define the norm ||f|| of a single function f on I to be the maximum of |f(x)| for x in the interval I. That is,

||f|| = max |f(x)| , x in I

The distance on an interval between two functions

Then the distance is given by

d_{I}(f_1, f_2) = || f_1 - f_2 || .

The corresponding notion of limit

If {f_n(x)} is a sequence of functions on the interval I, and f is also a function on I, then we make the following definition:

Definition

f = lim f_n

if

lim d_{I}(f, f_n) = 0 .


An example

f_n(t) = t^n e^{-nt},   I = [0, 1]

|| f_n || = e^{-n}


Exercise

Find || f_n ||_{I} when I = [0, 1] and f_n is the sequence of functions previously studied with (pointwise) limit 0 for which the corresponding sequence of integrals on the interval I has limit 1/2.
UP  |   TOP