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 norm1/2|| / ||x = ( √* 2 x ƒf²(x) dx) ¹/².a). Prove that X a Hilbert space and find the inner product which determines the abovenorm.b). Use the Gramm-Schmidt orthogonalization process on the set of functions f₁(x) =1, f₂(x) = x, ƒ3(x) = x².

As per the question we are given a functional space X along with its norm || ||X Now we have to :…

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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