I'm trying to prove exercise I.3.B in Mujica's "Complex analysis in Banach spaces".
DEFINITIONS:
A map $P$ is an m-homogeneous polynomial from $E$ to $F$ if there is a m-linear map $A$ from $E^m$ to $F$ such that $P(x)=A(x, \dots, x)$.
$P$ is a polynomial of degree at most $m$ if $P = P_0 + \dots + P_m$ where each $P_j$ is an j-homogeneous polynomial.
I have to find a function $f: E \to \mathbb{K}$ (where $E$ is infinite dimensional) such that $f(a + \lambda b)$ is a polynomial in $\lambda$ for all $a,b \in E$ but $f$ is not a polynomial.
$f$ clearly has to be discontinuous because there is a theorem implying that $f$ would be a polynomial in the continuous case.
I thought about considering something like (where $\theta$ stands for the step function):
$$f(a+\lambda b) = \theta (\| b\| -1) (a_1 + \lambda b_1)$$
But I don't know how to prove that $f$ wouldn't be a polynomial or even how to apply it to an arbitrary $x \in E$.
I also know that the restriction of $f$ to any finite dimensional subspace of $E$ is indeed a polynomial and that there is a sequence of homogeneous polynomials $P_k$ such that $f(x)=\sum_{k=0}^{\infty} P_k(x)$ where for each $x \in E$ $P_k(x)=0$ for all but finitely many indices.
What function could act as a good example for this situation?
I can provide any definition if you're not familiar with the terminology. Please ask for clarification in a comment if that is the case.
Let $\{x_i;i\in I\}$ be a Hamel basis for $E.$ If $E$ is infinite dimensional then without loss of generality we can have $I$ include the natural numbers as a subset.
Define
$$f:E\to\mathbb K:\sum_{i\in I}\alpha_ix_i\mapsto\sum_{n\in\mathbb N}\alpha_n^n.$$
Then $f$ cannot be a polynomial of degree at most $m$ for any $m,$ because for all $\lambda\in\mathbb K$ we have
$$f(\lambda x_{m+1})=\lambda^{m+1}$$
On the other hand for fixed $a$ and $b$ in $E$ the function $\lambda\mapsto f(a +\lambda b)$ is a polynomial in $\lambda$ because $a$ and $b$ are linear combinations of a finite number of elements from the Hamel basis.