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, || - |).
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 7E: For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={...
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![(a) 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 suci
that
x(1) = ao +aịt +... + aN-tN-1 + antN =Eajt,
for every r e [0, 1].
For any x e X written as above, consider X with the the norm given by
||x|| = max la l.
Os/SN
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
for every te [0, IJ.
=(1)"x
Show that (x,) is a Cauchy sequence in (X, || |I).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad4d02b6-78f1-4c0a-a713-3b04a8400050%2Fa3b69ac0-1f49-40c0-a917-14f82c4f3962%2Fj37ujm_processed.png&w=3840&q=75)
Transcribed Image Text:(a) 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 suci
that
x(1) = ao +aịt +... + aN-tN-1 + antN =Eajt,
for every r e [0, 1].
For any x e X written as above, consider X with the the norm given by
||x|| = max la l.
Os/SN
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
for every te [0, IJ.
=(1)"x
Show that (x,) is a Cauchy sequence in (X, || |I).
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