While solving an exercise I came across this problem
Let $V$ be a linear space. If $S:V→V$ and $T:V→V$ are functions which are commutative, that is, $ST=TS$ . Prove that
$(S+T)^2= S^2+2ST+T^2$
The way I attempted to solve it is If $x$ is an element of $V$ then
$(S+T)^2 (x)=(S+T)[(S+T)(x) ]=(S+T)[S(x)+T(x)]$
Had the functions been linear transformations then the last term of the above equation could have been written as
$S[S(x) ]+S[T(x) ]+T[S(x) ]+T[T(x) ]=(S^2+ST+TS+T^2 )(x)=(S^2+2ST+T^2 )(x)$
Since $S$ and $T$ are commutative.
But since the question nowhere mentions that the functions are linear transformations then is the hypothesis true for general functions as well as long as they are commutative?