Banach Space - Placement in The Hierarchy of Mathematical Structures

Placement in The Hierarchy of Mathematical Structures

Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.

The converse is not always true; not every Banach space is a Hilbert space. A necessary and sufficient condition for a Banach space X to be associated to an inner product (which will then necessarily make X into a Hilbert space) is the parallelogram identity:

for all x and y in X, and where is the norm on X. So, for example, while Rn is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm. Similarly, as an infinite-dimensional example, the Lebesgue space Lp is always a Banach space but is only a Hilbert space when p = 2.

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity. If X is a real Banach space, then the polarization identity is

whereas if X is a complex Banach space, then the polarization identity is given by (assuming that scalar product is linear in first argument):

The necessity of this condition follows easily from the properties of an inner product. To see that it is sufficient—that the parallelogram law implies that the form defined by the polarization identity is indeed a complete inner product—one verifies algebraically that this form is additive, whence it follows by induction that the form is linear over the integers and rationals. Then, since every real is the limit of some Cauchy sequence of rationals, the completeness of the norm extends the linearity to the whole real line. In the complex case, one can also check that the bilinear form is linear over i in one argument, and conjugate linear in the other.