Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

version of 27 August 1999

This appendix discusses some vector spaces which are *not*
inner product spaces, because
there is no way to define an inner product on them. All is not
necessarily lost! While we will not be able to talk about the angle
between vectors, it will still be possible to make sense of their
length (norm) and retain parts of the geometry we expect.
Let us look at a kind of vector spaces, called a
*sequence space*. This is essentially an ordinary
vector C^{n} with an infinite number n of components.
Thus a typical element would be **v** = (v_{1},
v_{2},...), except that the list of components never ends.
The information in the vector **v** is equivalent to that in
the sequence v_{k}. Now suppose that v_{k}
tends to zero for large k, but not very rapidly. It may happen that
is not always
finite but that
for some value
of p 1 other than 2.
The set of all such sequences is known as the sequence space
.

There is no inner product on when p 2, but there is still a norm.

Here we define the norm and verify that it functions properly. This
will require us to look at three inequalities, which are important in
analysis independently of the theory of vector spaces, *viz*.,
Young's,
Hölder's,
and
Minkowski's inequalities.

The first two of these use the concept of a dual index:

**Definition IIa.1**. If p is a number between 1 and infinity, then its *dual index* q is the number such than
1/p + 1/q = 1. A little simple calculation shows that q = p/(p-1).

**Theorem IIa.1** (Young's inequality). Suppose that x and y
are numbers and p and q are dual indices. Then

Proof. The proof is by elementary calculus. It is not hard to see that we may assume that x and y are positive, and this avoids some clutter. Fix y > 0, and find the maximum value of

xy - x^{p} /p

by differentiating with respect to x and setting the derivative to 0.
The maximizing value of x is y^{1/(p-1))}. We conclude that

xy - x^{p} /p
y^{p/(p-1)} - y^{p/(p-1)} /p,

which

y^{q} /q.

QED

Next suppose that **v** is in the sequence space
.
We claim that
we can define a norm for such
sequences by

.

The only property of the norm that is not obvious is the triangle inequality. If there is an inner product, the triangle inequality is a consequence of the CSB inequality, which is not true for all values of p. There is, however, a generalization of the CSB inequality:

**Theorem IIa.2** (Hölder's inequality). Suppose that **v** and **w**
are sequences in
and
,
and p and q are dual indices. Then

.

Proof. This follows from Young's inequality if we choose x and y very carefully. The good choices (again simplifying by supposing that everything is positive) are

.

This yields

If we then sum on the index k, the right side simplifies to 1/p + 1/q
= 1, and we get Hölder's inequality when we multiply by the
denominator on the left.

QED

Finally, we use this pretty much as in the familiar case where p=2
to derive a triangle inequality
for the *p-norm*:

**Theorem IIa.3** (Minkowski's inequality). Suppose that **v** and **w**
are sequences in
for p
1. Then

.

Proof. This is clear for p=1 because of the usual triangle inequality for the absolute value, so we suppose that p > 1. Then we write

and majorize the right side by

.

Now, on the right,

,

and similarly for the second term, except with ||**w**|| in place
of ||**v**|| - the other factor is identical. We then simplify by
recalling that (p-1)q = p and collect terms. Minkowski's inequality
falls out.

QED

We remark that there are integral versions of Hölder's and Minkowski's inequalities, as you might expect, so that if f is a function defined on a region , and

then

and ||f||_p defines a norm for all p 1.

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