Lecture 5, October 4, 2007

“Get-our-breath-back week”

I went back and worked through the proofs, more or less carefully, of the key results concerning closed linear subspaces; all about the existence and if possible also the uniqueness of a closest point inside such a subspace, to a given point outside. The final result about the decomposition of a Hilbert space into a sub-Hilbert space and its orthogonal complement uses indeed everything given in the assumptions: the parallelogram law for the norm is equivalent to existence of an inner product and is used to prove that a certain sequence is Cauchy; the completeness of the space tells us a limit point exists; the closure of the subspace shows that the limit point is inside the subspace.

Lecture 6, October 11, 2007

We do an extended version of R&Y 3.4 and 3.5.  Any Hilbert space has an orthonormal basis (a consequence of Zorn’s Lemma) and the cardinality of such a basis is well-defined, thus determining the isomorphism class of the Hilbert space. The Hilbert spaces with a countable orthonormal basis are precisely the separable ones. Each Hilbert space is isomorphic to an L₂-space for a counting measure.

An element of a Hilbert space can be developed as a generalized Fourier series in terms of an orthonormal basis, with only countably many terms being nonzero and the sum of the series being independent of the ordering of the terms. For L₂-spaces on the real line one has examples of orthonormal bases consisting of complex exponentials, sines or cosines.