n6 - ECE 580 / Math 587 SPRING 2011 Correspondence # 6...

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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....
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n6 - ECE 580 / Math 587 SPRING 2011 Correspondence # 6...

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