Conference on the Numerical Solution of Differential by J. L. Morris

By J. L. Morris

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CHAPTER 2. HILBERT SPACES 34 Proof: Let h ∈ H, define cj (h) = h, ej , and for n ∈ N let sn (h) = n j=1 cj ej ∈ H. 1: 0 ≤ ||h − sn (h)||2 n = h− n cj ej , h − j=1 n = ||h||2 − cj ej j=1 |cj |2 . j=1 This implies ∞. n 2 j=1 |cj | ≤ ||h||2 for every n and therefore ∞ 2 j=1 |cj | < We therefore obtain a linear map T : H → l2 (N) mapping h to the sequence (cj (h))j . Since nj=1 |cj (h)|2 ≤ ||h||2 we infer that ||T h|| ≤ ||h|| for every h ∈ H. For h in the span of (ej )j we furthermore have ||T h|| = ||h||.

Ek+1 are orthonormal. If H is finite-dimensional, this procedure will produce a basis (ej ) in finitely many steps and then stop. If H is infinite-dimensional, it will not stop and will thus produce a sequence (ej )j∈N . By construction we have span(ej )j = span(aj )j , which is dense in H. Therefore (ej )j∈N is an orthonormal basis. 2 Suppose H is an infinite-dimensional separable preHilbert space; and let (ej ) be an orthonormal basis of H. Then every element h of H can be represented in the form ∞ cj ej , h = j=1 where the sum is convergent in H, and the coefficients cj satisfy ∞ |cj |2 < ∞.

3. ORTHONORMAL BASES AND COMPLETION T✲ V H ❅ T 35 ❅ S ❅ ❘ ❄ H It is customary to consider a pre-Hilbert space as a subspace of its completion by identifying it with the image of the completion map. Again this theorem also holds for nonseparable spaces, but we prove it only for separable ones. Proof: Let V be a separable pre-Hilbert space. If V is finitedimensional, then V is itself a Hilbert space, and we can take T equal to the identity. 2. We have to show that T (V ) is dense in H = 2 (N). Let f ∈ 2 (N), and for n ∈ N let fn ∈ 2 (N) be given by fn (j) = f (j) if j ≤ n, 0 if j > n.

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