Then you probably learned that there are many more vector spaces than just

.

For example, the set of all polynomials, p(x), of degree n or less (for fixed n), is a vector space. We can define an "inner product" to be

.

More generally, the vector space,

, of functions, f(x), defined on [a, b] such that the (Lebesque) intgral

exists, is an infinite dimensional vector space. And we can define the inner product as [itex]<f, g>= \int_a^b f(x)g(x)dx[/tex]. The "length" of such a vector would be

, the set of all functions, f(x), defined on [a, b], such that

exists also forms a vector space but one on which we

**cannot** define an inner product on this space. We can define the length of a vector to be

.