Better to have more input variables and less output variables

 

When trying to describe an invariant map \phi : U \times V \to W, where $U,V$ and $W$ are finite-dimensional spaces,  \phi is “complicated” because its values are vector-valued. It is sometimes useful to look at \phi^{\sharp} : U \times V \times W^{*} \to {\mathbb K}, defined by \phi^{\sharp}(u,v,w^*)=w^*(\phi(u,v)). This map has more variables but takes scalar values.

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