Definition:Ordering on Multiindices

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Definition

Let $Z$ be the set of multiindices.


We define an ordering on $Z$ as follows.

Let:

$k = \sequence {k_j}_{j \mathop \in J}$
$\ell = \sequence {\ell_j}_{j \mathop \in J}$

be multiindices.


Then we write:

$k \preceq \ell$

if and only if:

$\paren {\forall j \in J} \paren {k_j \le \ell_j}$


By abuse of notation, it is standard to use $\le$ to denote the ordering on the set of multiindices as well.


Also see