Induced $\ell^2$ norm for a rank-three tensor

Question

Given an $n\times m$ matrix $A$, its $\ell^2$ induced norm is defined as $$ \|A\|_I = \sup_{x\in\mathbb{R}^m, x\neq 0} \frac{\|Ax\|_2}{\|x\|_2}. $$ Given a rank-three tensor $B$ with dimensions $n\times m\times p$ is it possible to define a similar induced norm?

Answer 1

In general, given a tensor $B\in V_1\otimes V_2\otimes V_3$, where $V_i$'s are all normed spaces (over $\mathbb R$ or $\mathbb C$, and if there is any problem with what follows, just assume they are all finite dimensional with an inner product), considered as multilinear map from $V_1^*,V_2^*, V_3^*$ to the scalar field, then we can define its norm by $$\|B\|:=\sup_{v_i\in V_i^*\setminus\{0\}}\frac{|B(v_1, v_2, v_3)|}{\|v_1\|\|v_2\|\|v_3\|}$$

Note that this definition generalized the original definition for norms on matrices, since $\|Ax\|_2=\sup_{y\in\mathbb R^n} \frac{|(Ax, y)|}{\|y\|}$

This definition can be generalized to tensors of any dimensions.

In particular, when $B$ is indeed a vector from a normed space $V$, then the defintion will not change the norm of $B$ on $V$, since we have the isometric embedding $V\hookrightarrow V^{**}$.