Equality of Polynomials

Definition
Let $(k, +, \circ)$ be an infinite field.

Let $k \left[\{ X_j : j \in J \}\right]$ be the ring of polynomial forms in the indeterminates $\{ X_j : j \in J \}$.

Let $f, g \in k \left[\{ X_j : j \in J \}\right]$

Then $f$ and $g$ are:


 * equal as functions if the polynomial functions associated to $f$ and $g$ are equal as functions, that is:


 * $\forall x\in k^J,\ f(x)=g(x)$

where $k^J$ is the free module on $J$.


 * equal as forms if the functions $M\to k$ from the free commutative monoid to $k$ which define $f$ and $g$ are equal as functions.

Theorem
$f$ and $g$ are equal as polynomials if and only if $f$ and $g$ are equal as functions.

Thus we can say $f = g$ without ambiguity as to what it means.