addition is defined by (f1 + f2)(x)=f1(x) + f2(x)
so, (f1 + f2)(-x)=f1(-x) + f2(-x) = f1(x) + f2(x)= (f1 + f2)(x). hence f1 + f2 belongs to U whenever f1 , f2 belongs to U.
similarly for multiplication.
show that U is a subspace of the given vetor space V.
V = F(R), ( the set of all functions from R into R) , U is the set of all even functions in V.
I am really stuck on this, if U is the set of all even functions in V, than x^2 must be in U? hence the set is non-empty, but i am not sure how to show that U is a subspace of V.
I know I have to show that if you take two elements from U and add them you should still get an element in U, but dont know how to go about it?
Any help appreciated.