Let $A_{ijk}$ be a symmetric tensor (meaning that $A_{ijk}=A_{jik}=A_{kij}=\cdots$). Is there a bound of the form
$$ |A_{ijk}| \leq C(d) \sup_{\|x\|_2=1} \left| \sum_{i,j,k=1}^d x_ix_jx_k A_{ijk}\right| ? $$
In other words, is the seminorm
$$ |A|_{pow} := \sup_{\|x\|_2 = 1} \left| \sum_{ijk} x_ix_jx_k A_{ijk} \right| $$
a norm when restricted to the subspace of symmetric tensors?
For the case of matrices this is a consequence of the polarization identity, $$ x^{\top} A y = \frac{1}{2} \big((x+y)^{\top} A (x+y) + (x-y)^{\top} A (x-y)\big). $$ Perhaps there is an analogous identity in the case of tensors? I am not sure because I know many facts about matrices do not generalize to tensors.