Theorem Let fbe a continuous function on an open subset of a Banach space X into a Banach space Y. Let a and b by any distinct points in U such that the segment [a, b] is contained in U and f be differentiable at each point of [a, b]. Let g be any continuous linear function on X into Y. Then ||f(b)-f(a)-g (b-a)| ≤c|| b-al, where c is a real number such that || Df (x)-g|| ≤c, for all x in [a, b].
Theorem Let fbe a continuous function on an open subset of a Banach space X into a Banach space Y. Let a and b by any distinct points in U such that the segment [a, b] is contained in U and f be differentiable at each point of [a, b]. Let g be any continuous linear function on X into Y. Then ||f(b)-f(a)-g (b-a)| ≤c|| b-al, where c is a real number such that || Df (x)-g|| ≤c, for all x in [a, b].
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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A1
![Theorem
Let f be a continuous function on an open subset U of a
Banach space X into a Banach space Y. Let a and b by any distinct points in U such
that the segment [a, b] is contained in U and f be differentiable at each point of
[a, b]. Let g be any continuous linear function on X into Y. Then
|| ƒ (b)-f(a) - g (b-a) || ≤c|| b-all,
where c is a real number such that
|| Df (x) - g || ≤c, for all x in [a, b].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff170082c-4fa4-4588-a272-ecf786b11ccc%2F7536cc02-88de-4c55-8e5e-af51a6b66e7f%2Fiyjbkl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Theorem
Let f be a continuous function on an open subset U of a
Banach space X into a Banach space Y. Let a and b by any distinct points in U such
that the segment [a, b] is contained in U and f be differentiable at each point of
[a, b]. Let g be any continuous linear function on X into Y. Then
|| ƒ (b)-f(a) - g (b-a) || ≤c|| b-all,
where c is a real number such that
|| Df (x) - g || ≤c, for all x in [a, b].
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