Definition:Ordering on Multiindices

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Let $Z$ be the set of multiindices.

We define an ordering on $Z$ as follows.

If $k = \left \langle {k_j}\right \rangle_{j \in J}$ and $\ell = \left \langle {\ell_j}\right \rangle_{j \in J}$ are multiindices, then we write $k \preceq \ell$ if:

$\left( \forall j \in J \right) \left( k_j \le \ell_j \right)$

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

Also see