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THEOREM 9.5. Let X be a Hilbert space and let A: X₁, X₂, ... be an orthonormal set in X. Then, if EF (n = 1, 2, ...), we have (1) Σ1%* converges if and only if Σ=₁ |α₂|² < ∞, and (2) if Σ₁%,* converges to x, then α = (x, xn). =1 Proof (2). Consider makking from *x to F x = 2 an n=1 S₂ = 24x₁. α; = (Sn, xj). Since this relationship is true (that is, the value remains the same) for any n> j, it must also be true in the limit, and we can write and the partial sum For n>j, we define Request explain the underlined parts Thm 9.2 это Xis an lim(S₁, xj) = αj. n mer product space, Appealing to Theorem 9.2, we can assert the continuity of the inner product mapping which allows us to interchange the operations of limit and inner product e inver product in the above equation, which yields _n,Y) is a continuous (x, xj) = αj, which is the desired result. Thus, knowing that the given series converges to some x, we have a strong relationship between the coefficients in the series and x.

THEOREM 9.5. Let X be a Hilbert space and let A: X₁, X₂, ... be an orthonormal set in X. Then, if EF (n = 1, 2, ...), we have (1) Σ1%* converges if and only if Σ=₁ |α₂|² < ∞, and (2) if Σ₁%,* converges to x, then α = (x, xn). =1 Proof (2). Consider makking from *x to F x = 2 an n=1 S₂ = 24x₁. α; = (Sn, xj). Since this relationship is true (that is, the value remains the same) for any n> j, it must also be true in the limit, and we can write and the partial sum For n>j, we define Request explain the underlined parts Thm 9.2 это Xis an lim(S₁, xj) = αj. n mer product space, Appealing to Theorem 9.2, we can assert the continuity of the inner product mapping which allows us to interchange the operations of limit and inner product e inver product in the above equation, which yields _n,Y) is a continuous (x, xj) = αj, which is the desired result. Thus, knowing that the given series converges to some x, we have a strong relationship between the coefficients in the series and x.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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THEOREM 9.5. Let X be a Hilbert space and let A: x₁, x₂,... be an orthonormal set in X. Then, if a F (n = 1, 2, ...), we have (1) Σ1%,* converges if and only if Σ%=₁ |α₁|² < ∞, and (2) if Σ₁ª* converges to x, then a, = (x, xn). Proof (2). Consider makking from XXX to F and the partial sum For n >j, we define x = Σ αnxn n=1 S₁₂ = Σ α₁x₁ dixi Request explain the underlined peerts αj = (Sn, xj). Since this relationship is true (that is, the value remains the same) for any n > j, it must also be true in the limit, and we can write Thm 9.2 90 X is an lim(Sn, xj) = α;. Inner product space, Appealing to Theorem 9.2, we can assert the continuity of the inner product mapping which allows us to interchange the operations of limit and inner product in the above equation, which yields the inner product (x,y) is a continuous (x, xj) = αj, which is the desired result. Thus, knowing that the given series converges to some x, we have a strong relationship between the coefficients in the series and x.
Transcribed Image Text:THEOREM 9.5. Let X be a Hilbert space and let A: x₁, x₂,... be an orthonormal set in X. Then, if a F (n = 1, 2, ...), we have (1) Σ1%,* converges if and only if Σ%=₁ |α₁|² < ∞, and (2) if Σ₁ª* converges to x, then a, = (x, xn). Proof (2). Consider makking from XXX to F and the partial sum For n >j, we define x = Σ αnxn n=1 S₁₂ = Σ α₁x₁ dixi Request explain the underlined peerts αj = (Sn, xj). Since this relationship is true (that is, the value remains the same) for any n > j, it must also be true in the limit, and we can write Thm 9.2 90 X is an lim(Sn, xj) = α;. Inner product space, Appealing to Theorem 9.2, we can assert the continuity of the inner product mapping which allows us to interchange the operations of limit and inner product in the above equation, which yields the inner product (x,y) is a continuous (x, xj) = αj, which is the desired result. Thus, knowing that the given series converges to some x, we have a strong relationship between the coefficients in the series and x.
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