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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 ϵ > , there exists d ∈ D such that ∥ x − d ∥ < ϵ . 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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 Spring '08
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 Linear Algebra, Vector Space, lim, Hilbert space

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