This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ECE 580 / Math 587 SPRING 2011 Correspondence # 6 February 21, 2011 Notes On HILBERT SPACES SEPARABILITY AND EXISTENCE OF BASIS I discuss in these notes the separability of Hilbert spaces, and in particular prove the important result that separability of a Hilbert space implies and is implied by the existence of a complete countable orthonor mal sequence (that is, a basis). This result is given in Theorems 1 and 2 below, which are followed by some discussion on the separability of L 2 ( , ). Let us first recall the notions of denseness and separability , which were introduced in class while dis cussing normed linear spaces. Definition 1. Given a normed linear space X , a subset D X is dense in X if for each x X and each > , there exists d D such that x d < . X is separable, if it contains a countable dense set. We now state and prove the two main theorems. Theorem 1. Let H be a separable Hilbert space. Then, every orthonormal system of vectors in H consists of a finite or a countable number of elements. Proof : Let { x 1 ,x 2 ,... } be a sequence of vectors dense in H , and let M be an orthonormal family in H . We need to show that M is countable. Let e 1 and e 2 be two distinct vectors in M . Choose x k 1 and x k 2 such that e 1 x k 1 < 1 2 2 and e 2 x k 2 < 1 2 2 . By orthonormality, e 1 e 2 2 = e 1 2 + e 2 2 = 2 , and hence 2 = e 1 e 2 e 1 x k 1 + e 2 x k 1 < 1 2 2 + e 2 x k 1 e 2 x k 1 > 1 2 2 Hence x k 1 = x k 2 and k 1 = k 2 . Thus, we can associate with each element of M a different integer k , which shows that M is enumerable (that is, countable). The next result says that separability is actually equivalent to existence of a complete orthonormal sequence. Theorem 2. A Hilbert space H contains a complete orthonormal sequence if and only if it is separable....
View
Full
Document
 Spring '08
 Staff

Click to edit the document details