Question

consider the set of all continuous functions with operations of pointwise addition and scalar multiplication, having domain [0, 1]. is this set a vector space? explain.

Answer 1

check the axioms one at a time
that is the point of this exercise

Answer 2

is it clear what you have to check?

Answer 3

i am some what confused on what to do since it is a domain and not a vector

Answer 4

not sure what exactly you mean "the domain is not a vector"
the continuous functions are the vectors

Answer 5

you have to check the axioms for a vector space:
\(x+y=y+x\) for example where in this case \(f+g=g+f\) where \(f, g\) are functions continuous on \([0,1]\)

Answer 6

this is true because by definition \((f+g)(x)=f(x)+g(x)=g(x)+f(x)=(g+f)(x)\)

Answer 7

you also have to check that \(0\in V\) which it is, it is the zero function i.e. \(0(x)=0\) for all \(x\) is continuous on \([0,1]\)

Answer 8

most of what you need is inherited from the properties of real numbers, and the fact that addition is point wise. check all the axioms one at a time, you will see it