This is a functional analysis problem, i need solutions for all 3 subpoints, but if you cannot solve them all it is alright to solve at least the first subpoint. Note that in the first subpoint you should prove the all 5 axioms for linear space, thanks in advance. Problem is in the attached picture. Consider [a,b] c R, we define the set:C([a,b],R") = {f : [a,b] → R", f=(f1,..,fn)/ f is continous on [a,b]}Show that the following statements are true:i)C([a,b],R") forms a linear space in relation to the estimation operations of two functions in C([a,b],R"),respectively of multiplication with real scalars with functions from C([a,b],R").ii) The map ||.||c : C([a,b],R") → R,IIfle = max;=1,n maxx¤[a,b] lfi (x)| define a norm on space C([a,b],R") named Cebisev norm.iii) Normed linear space C([a,b],R"), |l-lI.) is complete

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Description: This is a functional analysis problem, i need solutions for all 3 subpoints, but if you cannot solve them all it is alright to solve at least the first subpoint. Note that in the first subpoint you should prove the all 5 axioms for linear space, than

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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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This is a functional analysis problem, i need solutions for all 3 subpoints, but if you cannot solve them all it is alright to solve at least the first subpoint. Note that in the first subpoint you should prove the all 5 axioms for linear space, thanks in advance. Problem is in the attached picture.

Consider [a,b] c R, we define the set:
C([a,b],R") = {f : [a,b] → R", f=(f1,..,fn)/ f is continous on [a,b]}
Show that the following statements are true:
i)C([a,b],R") forms a linear space in relation to the estimation operations of two functions in C([a,b],R"),
respectively of multiplication with real scalars with functions from C([a,b],R").
ii) The map ||.||c : C([a,b],R") → R,
IIfle = max;=1,n maxx¤[a,b] lfi (x)| define a norm on space C([a,b],R") named Cebisev norm.
iii) Normed linear space C([a,b],R"), |l-lI.) is complete

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