This bilinear map can be described in terms of a set of " connection coefficients " ( also known as Christoffel symbols ) specifying what happens to components of basis vectors under infinitesimal parallel transport:
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In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.
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We will let \ mathcal { B } ( X, Y; Z ) denote the space of separately continuous bilinear maps and B ( X, Y; Z ) denote its subspace the space of continuous bilinear maps, where X, Y and Z are topological vector space over the same field ( either the real or complex numbers ).
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We will let \ mathcal { B } ( X, Y; Z ) denote the space of separately continuous bilinear maps and B ( X, Y; Z ) denote its subspace the space of continuous bilinear maps, where X, Y and Z are topological vector space over the same field ( either the real or complex numbers ).
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For instance, " linear algebra duality " corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the " duality between distributions and the associated test functions " corresponds to the pairing in which one integrates a distribution against a test function, and " Poincar?duality " corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.
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More abstractly, the outer product is the bilinear map W \ times V ^ * \ to \ operatorname { Hom } ( V, W ) sending a vector and a covector to a rank 1 linear transformation ( simple tensor of type ( 1, 1 ) ), while the inner product is the bilinear evaluation map V ^ * \ times V \ to F given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector / vector distinction.
