Let $C$ be a three-dimensional tensor of dimensions $n\times n\times n$. Define: $[C(x,y)]_k=\sum_{i,j}C_{ijk}x_iy_j,$ i.e. $C(x,y)$ is a vector of dimension $n$.
Is there a way to bound the norm: $||C(x,y)||_2$ such that the bound depends on some norm of $C$ and some norms of $x$ and $y$? (something like a generalized Cauchy-Schwarz).
What about a similar bound on $||C(x,y)-C(x',y')||_2$ (something that will depend on $||C||$ and say $||x-x'||$ and $||y-y'||$)?
Thanks.