THÉOREM 9.8. Let X be a Hilbert space, suppose that A = {x}, is an ortho-normal set in X, and let x be an arbitrary vector in X. Then the following statementsare true.(1) y = [₂€ A(x, x₂)x₁ exists; that is, the series is summable;(2) the vector y mentioned in (1) belongs to [A];(3) x = [A] if and only if x = y (it can be written as the above series);(4) x − y ¹ [A].Proof (1). We note that Σ(x, x₂)x² converges, sinceΣ ||(x, x₂)x₂||² = Σ |(x, x,)|² ≤ ||x||²by the Bessel inequality [Theorem 9.3(1)].RequestexplainProof for (1) continuesProof (2). It is clear that any partial sum of (x, x)x, must belong to [4]. Thisimplies that the limit y must belong to [4].Request clocity proof (2)

note : As per the instructions we will only solve the part (1), (2) of the question.

Answered: THÉOREM 9.8. Let X be a Hilbert space,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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THÉOREM 9.8. Let X be a Hilbert space, suppose that A = {X₂}, is an ortho- normal set in X, and let x be an arbitrary vector in X. Then the following statements are true. (1) y = [₂€ A(X, x₂)x, exists; that is, the series is summable; (2) the vector y mentioned in (1) belongs to [A];