Definition of Hilbert space

A Hilbert space has the same relationship to an inner product space as a Banach space has to a normed space. Since any inner product space is a normed space, the norm being determined by the inner product as in Section 8.1, nothing new is involved in the notion of completeness for an inner product space.

Definition 9.1.1 A complete inner product space is called a Hilbert space.

It follows that every Hilbert space is also a Banach space, though the converse is certainly not true: C [a, b] is a Banach space but not a Hilbert space, since it is not even an inner product space. Any finitedimensional inner product space is complete (Theorem 6.5.4), so all finite-dimensional inner product spaces are Hilbert spaces.

As a metric space, we have seen that C2[a, b] is not complete, and so it cannot be complete as an inner product space. That is, C2[a, b] is not a Hilbert space. As our main example of a Hilbert space, except for finitedimensional ones like Cn, we are thus left with the space l2. There is an important analogue of the space C2[a, b], where the integral for the space is developed in a different manner from the usual (Riemann) integral, so that, as one consequence, the space turns out to be complete. The Lebesgue integral is an example of such an integral, but its treatment is beyond the scope of this book.

However, much of what follows will be valid for inner product spaces in general, and, although we have only one example here of an infinitedimensional Hilbert space, there is plenty to discuss just with regard to l2.