Codimension of the image of the polynomials subspace is infinite

Question

Consider the interval $I=[0,1]$ and the Banach space $E$ of real continuous functions defined on $I$ ($E=\mathcal C_{\mathbb R}(I))$.

$P \subset E$ is the subspace of polynomial functions (restricted to $I$). Take $0 < \epsilon < 1$ and define $f: P \mapsto P$ by $f=u+\epsilon v$ where $u$ is the identity and $v: x(t) \to x(t^2)$.

How to prove that the codimension of $f(P)$ in $P$ is infinite?

Answer 1

Hint: for any $p\in P$, the degree of $f(p)$ is even.