norm of three dimensional matrices I need to use the Lipschitz continuity of a three-dimensional-matrix-valued function $A = (A_1,\dots,A_n):\mathbb{R}^n\to\mathbb{R}^{l\times m\times n}$ where $A_i:\mathbb{R}^n\to\mathbb{R}^{l\times m}$. So, I need to define a norm of three dimensional matrices. I may define it as $\|A\| = \sum_{i=1}^n\|A_i\|$ with a norm of $\mathbb{R}^{l\times m}$. Is it a typical choice? Are there well-defined norms in $\mathbb{R}^{l\times m\times n}$ as the norms in $\mathbb{R}^{l\times m}$? Does this definition inherits the properties of the norm in $\mathbb{R}^{l\times m}$ used? Lastly, what is the proper subject to find the information needed?

$\Bbb R^{l\times m \times n} \equiv \Bbb R^{lmn}$. If $x = (x_1,...,x_{lmn}) \in \Bbb R^{lmn}$, you might recall that a common norm for the latter space is $$\|x\|_2 =\sqrt{\sum_{i=1}^{lmn} x_i^2}$$
Other common choices include
$$\|x\|_1 = \sum_{i=1}^{lmn} |x_i|$$
and $$\|x\|_0 = \max \{|x_1|, |x_2|, \dots, |x_{lmn}|\}$$
The thing is, all of these norms are equivalent. They give the same topology (a set that is open under one is also open under the others), and they give the same concept of Lipschitz continuity: a map that is Lipschitz continuous for one is Lipschitz continuous for the others. 
Depending on which matrix norm you mean by $\|A_i\|$, your norm may be a cross between $\|\cdot\|_1$ and $\|\cdot\|_2$. In any case, I don't doubt that it is also equivalent to the others.
So you are free to pick any of them.
