Determine whether the set of all odd functions fx fx is a s

Determine whether the set of all odd functions f(-x) = -f(x) is a subspace of all functions of the type C(-infinity, +infinity)

Solution

to show that set of all odd functions is a subspace of all functions we just need to show that af and (f+g) is in C where a is some real number and f,g are odd functions

then f(-x)=-f(x) and g(-x)=-g(x) by definition of odd function

then (f+g)(-x)=f(-x)+g(-x)=-f(x)-g(-x)=-(f+g)(x)

Hence (f+g) belongs to C

now af(-x)=a(-f(x))=-af(x)
Hence af belongs to C

Hence it is proved that set of all odd functions is a subspace of all functions.