Examples of convergence

Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

#### *(c) Copyright 1994,1997,2000 by Evans M. Harrell
II and James V. Herod. All rights reserved.

version of 23 February 2000

Here are two examples illustrating different ways in which a
sequence of functions can converge.
To illustrate these ideas, two examples follow. In the first, there is a
sequence {f_{p}} and a function g in the space with
lim_{n} f_{n} = g. In the second, there is no such g.

**Example C.1**: Let E be the vector space of continuous functions on [0,1]
with the usual
inner product. Let

and let g(x) = x on [0,1]. Then lim_{n} I|f_{n} - g||^{2}

As in
Chapter II, we say that f_{n} converges
in the
root-mean-square, or L^{2}, sense.

**Example C.2**. This space E of continuous functions on [0,1]
with the "usual"
inner product is not complete. To establish this, we provide a sequence
f_{p} for which there is no __continuous__ function g
such that lim_{n} f_{n} = g.

Sketch the graphs of f_{1}, f_{2}, and f_{3} to see that the limit of this sequence of
functions is not continuous.
In
chapter III
we introduced the notion of
uniform convergence. Recall:

**Definition III.7**.
A sequence of functions {f_{k}(x)} converges *uniformly* on
the set to a function g provided that

|f_{k}(**x**) - g(**x**)| < c_{k},
(*)
where c_{k} is a sequence of constants (independent of **x** in
) tending to 0.
Some terminology you may encounter for condition (*) is that
the sequence f_{k}(**x**) is *Cauchy* in the
uniform sense. This condition guarantees that it converges
uniformly to a limit, and that the limit is continuous:

**Theorem**.
Suppose that **x** ranges over a closed, bounded set
. If m > n implies that

|f_{n}(**x**) - f_{m}(**x**) | < c_{n},
where the constants (independent of **x**) c_{n}
0 as n
0,
then there is a continuous function g(**x**) such that
f_{k}(**x**) g(**x**)
uniformly on .

Link to
chapter II
chapter III
chapter XV
Table of Contents
Evans Harrell's home page
Jim Herod's home page