Let I be a bounded interval, and Z a Banach space. Let (fn) be a sequence of differentiable functionsfn : I → Z such that -1 fn(x°) converges to Z for some x° E I and there exists a sequence of positivenumbers (Mn) such that || f,(x)|| < M, for all n E N and x E I and E-1 Mk converges in R.Show that , fn (x) converges uniformly for x € I and provides a differentiable function f : I → Zsuch that f'(x) = E1 fn(x), with the convergence of the latter series being also uniform for x E I.n=1n=1n=1n=1

Name: Let I be a bounded interval, and Z a Banach space. Let (fn) be a sequ
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Description: Let I be a bounded interval, and Z a Banach space. Let (fn) be a sequence of differentiable functionsfn : I → Z such that -1 fn(x°) converges to Z for some x° E I and there exists a sequence of positivenumbers (Mn) such that || f,(x)|| < M, for all

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let I be a bounded interval, and Z a Banach space. Let (fn) be a sequence of differentiable functions
fn : I → Z such that -1 fn(x°) converges to Z for some x° E I and there exists a sequence of positive
numbers (Mn) such that || f,(x)|| < M, for all n E N and x E I and E-1 Mk converges in R.
Show that , fn (x) converges uniformly for x € I and provides a differentiable function f : I → Z
such that f'(x) = E1 fn(x), with the convergence of the latter series being also uniform for x E I.
n=1
n=1
n=1
n=1

Expert Solution

Step 1

given

let $I$ be a bounded interval and $Z$ a banach space . let $f_{n}$ be a sequence of differentiable functions $f_{n} : I \to Z$ such that $\sum_{n = 1}^{\infty} f_{n} (x^{⋄})$ converges to $Z$ for some $x^{⋄} \in I$ and there exist a sequence of positive numbers $(M_{n})$ such that $||f_{n}^{'} (x)|| \leq M_{n} \forall x \in I and \sum_{n = 1}^{\infty} M_{k} converges in ℝ$

to show

$\sum_{n = 1}^{\infty} f_{n} (x) converges uniformly for x \in I$ and provides a differentiable function $f : I \to Z$ such that $f^{'} (x) = \sum_{n = 1}^{\infty} f_{n}^{'} (x)$

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