The Clifford algebra of the dual space

An immediately apparent notational issue with the Clifford algebra of \({V^{*}}\) is that juxtaposition is used to denote both Clifford multiplication and the multiplication of the scalar values of forms. However, they can be distinguished if the arguments are explicitly noted, e.g. the Clifford product of 1-forms is written \({\varphi\psi\left(v,w\right)}\) versus the scalar product of values \({\varphi\left(v\right)\psi\left(w\right)}\).

We can use the view of exterior forms as mappings to view the Clifford product of 1-forms as \({\varphi\psi\left(v,w\right)=\left\langle \varphi,\psi\right\rangle +\left(\varphi\wedge\psi\right)\left(v,w\right)\mapsto\left\langle \varphi,\psi\right\rangle +\varphi\left(v\right)\psi\left(w\right)-\psi\left(v\right)\varphi\left(w\right)}\). Note that in this view the Clifford product is not a multilinear mapping on \({V}\), since there is a leading constant; it is an affine mapping (defined here). The Clifford multiplication of higher-grade \({j}\)- and \({k}\)-vectors yields a constant plus multivectors of mixed grade, which in this view are then a sum of multilinear mappings on subsets of the \({\left(j+k\right)}\) vector arguments.

An Illustrated Handbook