# is there a way to bound the following 2-norm?

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.

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We have $$[C(x,y)]_k^2\leq \left(\sum_{i,j}C_{ijk}^2\right)\left(\sum_{i,j}x_i^2y_j^2\right)=\lVert x\rVert_2^2\cdot\lVert y\rVert_2^2\sum_{i,j}C_{ijk}^2$$ so putting $\lVert C\rVert_2^2:=\sum_{i,j,k}C_{ijk}^2$ we get $\lVert C(x,y)\rVert\leq\lVert C\rVert_2\cdot \lVert x\rVert_2\cdot \lVert y\rVert_2$. For a fix $x$, the map $y\mapsto C(x,y)$ is linear hence \begin{align} \lVert C(x,y)-C(x',y')\rVert_2&\leq\lVert C(x,y-y')\rVert_2+\lVert C(x-x',y'\rVert_2\\ &\leq \lVert C\rVert_2\lVert x\rVert_2\lVert y-y'\rVert_2+\lVert C\rVert_2\lVert x-x'\rVert_2\lVert y'\rVert_2\\ &\leq \lVert C\rVert_2\max\{\lVert x\rVert_2,\lVert y'\rVert_2\} \left(\lVert x-x'\rVert_2+\lVert y-y'\rVert_2\right). \end{align}