C[[a,b],R") = {f : [a,b] → R", f=(f,...,f)/ 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 respectively of multiplication with real scalars with functions from C([a,b],R"). ii) The map |.lle : C([a,b],R") → R, Ifle = 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|.) is complete
C[[a,b],R") = {f : [a,b] → R", f=(f,...,f)/ 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 respectively of multiplication with real scalars with functions from C([a,b],R"). ii) The map |.lle : C([a,b],R") → R, Ifle = 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|.) is complete
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a2b3a39-4fa1-46ba-8703-d52c5b98ac34%2F62101e20-fb55-4622-81e2-a1cbfc95e9a3%2Fxejfzqm_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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