Consider $E$ and $F$ two vector spaces over $\mathbb{R}$, and $f : E^n \longmapsto F$ a $n$-linear map.
Assume that $f$ is symmetric, i.e. for all $(v_1,...,v_n) \in E^n$, for all permutation $\sigma \in S_n$, $$f(v_1,...,v_n) = f(v_{\sigma(1)},...,v_{\sigma_n}).$$ Then show that $f$ is uniquely defined by the values of the $f(v,v,...,v)$ for $v \in E$.
I tried showing the result using induction, but had some problems deriving $n+1$ from $n$. I mainly used the following base case.
Proof for $\textbf{n=2}$: for all $(u, v) \in E^2$, denote $a = \frac{u+v}{2}$ and $b= \frac{u-v}{2}$. Then the multilinearity yields $f(u,v) = f(a+b,a-b) = f(a,a)+f(b,a)-f(a,b)-f(b,b)$ and as $f$ is symmetric, $f(u,v) = f(a,a)-f(b,b)$.
