Let X be the space of all real valued polynomials of any degree in the closed interva [0, 1]. That is, x EX if and only if there exist N e N and ao, a1.....aN ER suc that x(t) = ao + aịt +,.. + aN-it-1 + antN = for every t e [0, 1]. For any x e X written as above, consider X with the the norm given by || || = max la l. OsjSN That is, the norm of x is equal to the maximum of the absolute value of the coeffi- cients a, j = 1,...,N. i. Let = (1)"x j! for every te [0, 1]. Show that (x,) is a Cauchy sequence in (X, || - |).
Let X be the space of all real valued polynomials of any degree in the closed interva [0, 1]. That is, x EX if and only if there exist N e N and ao, a1.....aN ER suc that x(t) = ao + aịt +,.. + aN-it-1 + antN = for every t e [0, 1]. For any x e X written as above, consider X with the the norm given by || || = max la l. OsjSN That is, the norm of x is equal to the maximum of the absolute value of the coeffi- cients a, j = 1,...,N. i. Let = (1)"x j! for every te [0, 1]. Show that (x,) is a Cauchy sequence in (X, || - |).
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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