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...

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.

Extracted 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