# About the symmetric multilinear maps

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, $$f(u,v) = f(a,a)-f(b,b)$$.

• I suppose you need to assume the field is not of characteristic $2$, and probably not of any finite characteristic either. – edm Sep 24 '18 at 12:22
• @edm that's right ! I am going to define the space vectors over R – charmd Sep 24 '18 at 12:41
• – Jens Schwaiger Sep 24 '18 at 13:09