Operator norms of symmetric multilinear maps on $(\mathbb{R}^k)^n$

Question

Let $T:(\mathbb{R}^k)^n \rightarrow \mathbb{R}$ be a (continuous) symmetric $n$-multilinear map and $M>0$.

Assume that $|T(x,...,x)|\leq M$ for each $x\in \mathbb{R}^k$ such that $||x||=1$.

Then, how do I show that $|T(x_1,...,x_n)|\leq M||x_1||...||x_n||$ for each $x\in (\mathbb{R}^k)^n$?

Answer 1

The question has been already aksed in https://mathoverflow.net/questions/207998/norm-of-n-linear-symmetric-forms/208025#208025?newreg=8d0314c729f840789c615a033e5fad50

The proof seems difficult ...