I have been reading through Linear Algebra Done Right by Sheldon Axler. The book defines an operator as a linear map from a vector space to itself. It then considers at another part of the book the operator of differentiation and states that the derivative of any polynomial of degree at most $4$ is a polynomial of degree at most $4$. I assume this means that this set polynomials is invariant under the differential operator. My crux with this is that wouldn't the differential operator map the vector space of polynomials of at most degree $4$ to the set of polynomials of at most degree $3$?
Apologies if this question does not make sense.