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
dI(f1, f2) =
|| f1 - f2 || .
The corresponding notion of limit
If {fn(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 fn
if
lim dI(f, fn) = 0 .
An example
fn(t) = tn e-nt, I = [0, 1]
|| fn || = e-n
Exercise
Find || fn ||I when I = [0, 1]
and fn 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.
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