The interior product is a dual notion of the wedge product in an exterior algebra , where is a vector space. Given an orthonormal basis of , the forms
(1)
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are an orthonormal basis for . They define a metric on the exterior algebra, . The interior product with a form is the adjoint of the wedge product with . That is,
(2)
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for all . For example,
(3)
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and
(4)
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where the are orthonormal, are two interior products.
An inner product on gives an isomorphism with the dual vector space . The interior product is the composition of this isomorphism with tensor contraction.