Consider [a,b] c R, we define the set: C([a,b],R") = {f : [a,b] → R", f=(f,..f)/ f is continous on [a,b]} Show that the following statement is true: C([a,b],R") forms a linear space in relation to the...

Functional analysis and Linear algebra problem!

Problem is in the attached picture.

Show that Real valued continuous functions on [a,b] are vector space, but in Rⁿ
Space. please check the attached image.

I know that we have to prove the 10 axioms for vector space but i don't know how to apply this axioms here on Rⁿ
Space.

Extracted text: Consider [a,b] c R, we define the set: C([a,b],R") = {f : [a,b] → R", f=(f,..f)/ f is continous on [a,b]} Show that the following statement is true: 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").