Lecture 4, September 27, 2007

We do the paragraphs R&Y 3.1, 3.2 and 3.3. Starting with the basic definitions of an inner product space, the Cauchy-Schwarz inequality, and the associated norm, we defined the notion of a Hilbert space. “The only” example of a Hilbert space, namely L ² (X), was explained.  Rⁿ, Cⁿ, l ², and of course L ² [0,1], are all particular cases of this universal model (which was also a good excuse to mention the Hölder inequalities from R&Y 1.54).

Subspaces and products of inner product spaces carry a natural inner product.

A particular feature of a norm which comes from an inner product is that it satisfies the parallellogram. Although we do not prove this, it is a fact that a norm which satisfies this rule comes from an inner product. How one should define this product is made obvious by the so-called polarization identities, expressing an inner product in terms of norms.

One has the usual notions of orthogonality, orthonormality and the Gram-Schmidt procedure, familiar from linear algebra. Things start to get interesting in the geometrical Theorem 3.32, stating that the distance between a closed convex subset of a Hilbert space and a point is always realized by a unique point of the subset. This is the basis for the workhorse of the subject, Theorem 3.34, which shows that, given a closed subspace Y of a Hilbert space, the whole space has an orthogonal decomposition into Y and its orthoplement. Such a decomposition result does not hold for general Banach spaces: there are examples of a Banach space X and a closed subspace Y where there does not exist a closed subspace Z such that X is the direct sum of Y and Z.

Closely related to the decomposition theorem is the description in Corollary 3.36 of the double orthocomplement of any subspace of a Hilbert space: you simply get the closure of what you started with.

Read R&Y 3.1–3.3. The exercises don’t seem to be particularly challenging, with perhaps 3.20 on page 72 as an exception. Part (a) shows that Corollary 3.36 holds for arbitrary inner product spaces and part (b) is a logical test.