# 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.

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