Operations On Super Vector Spaces
The dual space V* of a super vector space V can be regarded as a super vector space by taking the even functionals to be those that vanish on V1 and the odd functionals to be those that vanish on V0. Equivalently, one can define V* to be the space of linear maps from V to K1|0 (the base field K thought of as a purely even super vector space) with the gradation given in the previous section.
Direct sums of super vector spaces are constructed as in the ungraded case with the grading given by
One can also construct tensor products of super vector spaces. Here the additive structure of Z2 comes into play. The underlying space is as in the ungraded case with the grading given by
where the indices are in Z2. Specifically, one has
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