Geometry without an Inner Product

Linear Methods of Applied Mathematics Evans M. Harrell II and James V. Herod*

version of 27 August 1999

Geometry without an Inner Product.

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 Cn with an infinite number n of components. Thus a typical element would be v = (v1, v2,...), except that the list of components never ends. The information in the vector v is equivalent to that in the sequence vk. Now suppose that vk 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 .

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 - xp /p

by differentiating with respect to x and setting the derivative to 0. The maximizing value of x is y1/(p-1)). We conclude that

xy - xp /p yp/(p-1) - yp/(p-1) /p,
which
yq /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.