2. Consider the following space of measurable functions defined on [0,1]: X = {ƒ : [ ² zf²(x)dx < ∞ } equipped with the norm 1/2 || S || x = (S*' 2x5²(x)dx) ¹². a). Prove that X a Hilbert space and find the inner product which determines the above norm. = b). Use the Gramm-Schmidt orthogonalization process on the set of functions f₁(x) = 1, f₂(x) = x, f3(x) = x².
2. Consider the following space of measurable functions defined on [0,1]: X = {ƒ : [ ² zf²(x)dx < ∞ } equipped with the norm 1/2 || S || x = (S*' 2x5²(x)dx) ¹². a). Prove that X a Hilbert space and find the inner product which determines the above norm. = b). Use the Gramm-Schmidt orthogonalization process on the set of functions f₁(x) = 1, f₂(x) = x, f3(x) = x².
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the following space of measurable functions defined on [0,1]: equipped with the norm "!1 # X = f : xf2(x)dx < ∞ 0 $! 1 %1/2 " f "X= 2xf2(x)dx . 0 a). Prove that X a Hilbert space and find the inner product which determines the above norm. b). Use the Gramm-Schmidt orthogonalization process on the set of functions f1(x) = 1, f2(x) = x, f3(x) = x2.
![2. Consider the following space of measurable functions defined on [0,1]:
X = {1 : [ ' xf² (x) dx < ∞ }
equipped with the norm
1/2
|| / ||x = ( √* 2 x ƒf²(x) dx) ¹/².
a). Prove that X a Hilbert space and find the inner product which determines the above
norm.
b). Use the Gramm-Schmidt orthogonalization process on the set of functions f₁(x) =
1, f₂(x) = x, ƒ3(x) = x².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc7a159ee-ed17-4137-bfca-c0c828eb7e35%2F9d33087f-81ad-46e6-8325-e7d4d85836e1%2Fgmkdcm6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Consider the following space of measurable functions defined on [0,1]:
X = {1 : [ ' xf² (x) dx < ∞ }
equipped with the norm
1/2
|| / ||x = ( √* 2 x ƒf²(x) dx) ¹/².
a). Prove that X a Hilbert space and find the inner product which determines the above
norm.
b). Use the Gramm-Schmidt orthogonalization process on the set of functions f₁(x) =
1, f₂(x) = x, ƒ3(x) = x².
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