James V. Herod*

Page maintained with additions by Evans M. Harrell, II, harrell@math.gatech.edu.

**SECTION1. MORE ABOUT FUNCTION SPACE**

This section will continue to use the basic idea that a collection of functions defined on an interval might form a vector space and that the vector space of functions might have an inner product defined on it.

SECTION 2.1 MORE ABOUT L^{2}[0,1]

Recall that L^{2}[0,1] with the "usual" inner product is such an
inner product space of functions on [0,1]. (See
Chapter B2 for more discussion of inner products.)
One should be aware that there are
many choices that can be made for an inner product for the functions on [0,1].
One might have a weighted inner product such as we had in R^{n} and as
was described in Chapter 1. That is, if w(x) is a continuous, non-negative
function, we might have

The choices for w(x) are usually suggested by the context in which the space arises.

Some inner product spaces are called HILBERT SPACES. A
Hilbert
space is
simply a vector space on which there is an inner product and on which there is
one more bit of structure. A
Hilbert
space is a *complete* inner product
space. This means that if f_{p} is an infinite sequence of
vectors in the space which is
Cauchy
convergent - meaning that

lim_{n,m} < f_{n}-f_{m} , f_{n}-f_{m} > = lim_{n,m} | f_{n}-f_{m} |^{2} = 0

- then there is a vector g, also in the space, such that lim_{n} f_{n} = g.

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**: 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} |f_{n} - g|^{2}

**EXAMPLE**. 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.

The
Riesz
Representation Theorem is an important idea in a
Hilbert
space. You
will recall that in R^{n}, this result declared that if L is a linear
function from R^{n} to R then there is a vector **v **in
R^{n} such that L(x) = < x, **v **> for each x in
R^{n}.
(More description of the analogue of
Riesz representation for R^{n}.)

In general, where the vector space is not R^{n}, more
is required.

**THEOREM** If {E, < , >} is a
Hilbert
space, then these are
equivalent:

(a) L is a continuous, linear function from E to R (or the complex numbers C), and

(b) there is a member **v** of E such that L(x) = < x, **v **> for
each x in E.

It is not hard to show that (b) implies (a). To show that (a) implies (b) is more interesting. The argument uses the fact that E is complete and can be found in any introduction to Hilbert space or functional analysis.

**EXAMPLE**. Since the space of continuous functions on [0,1], denoted
C[O,1], with the usual inner product is __not__ a
Hilbert
space, one should
expect that there might be a linear function L from C[0,1] to R for which there
is no **v** in C[0,1] such that

for each f in C[0,1]. In fact, here is such an L:

Let

The candidate for **v** is **v**(x) =1 on [3/4, 1] and 0 on [0, 3/4).
But this **v** is not continuous! It is only piecewise continuous.

**DEFINITION.** The
Heaviside function **H** is defined as follows:

(2.1)Note that

Thus, the Heaviside function provides an element **v** for which the linear
function

has a
Riesz representation. As noted, **v** is not in C[0,1].

It is not always possible to have a piecewise continuous **v** which will
rectify the situation. Consider the following linear function: L(f) = f(1/2).
It is not so hard to see that there is no piecewise continuous function
**v** on [0,1] having the property that for every continuous f,

**DEFINITION**. The symbol \delta is used to denote the "generalized" function
which has the property that

(2.2)

for some suitably large class of functions f, when 0 < a < 1. It is no surprise that some
effort has been made to develop a theory of *generalized functions* in
which the delta function can be found. Generalized functions are also called
"distributions". While the delta function is not a well-defined function of
the familiar type, it can be manipulated like a function in most cases,
provided that in the end it will be integrated, and that the other quantities
in the integral with it will be continuous.

A suggestive analogy is that of complex numbers. Like complex numbers, generalized functions are idealizations which do not describe real, physical things, but they have been found tremendously useful in applied mathematics. Most famously they were used by Dirac in his quantum mechanics; it is less well known that mathematical physicist Kirchhoff used them decades earlier in his work on optics. The theory of distributions is attractive and establishes a precise basis for the ideas which these notes will use. It is the choice of this course and these notes, however, to use the delta function without exploring the mathematical framework in which it should be studied.

We shall return to the delta function when its properties are needed to understand how to construct Green functions.

**EXERCISES 2.1: **

1. Calculate the following integrals:

2. Let <f,g> denote the standard inner product for 0 <= x <= 1.

Find a generalized function g(x) such that

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