If is a real vector space, then we replace by its complexification and let denote the induced bilinear form on.
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So, an inner product on a real vector space is a " positive-definite symmetric bilinear form ".
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When the scalar field is the real numbers, the vector space is called a " real vector space ".
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They have applications in algebraic topology, where they are used to define lattice theory studies free abelian subgroups of real vector spaces.
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Vect "'C "', from the category of real vector spaces to the category of complex vector spaces.
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A bilinear form on a real vector space " V ",, corresponds in a natural way to a type tensor in
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In the language of category theory one says that complexification defines an ( category of real vector spaces to the category of complex vector spaces.
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Doing this constructions pointwise gives the tangent space, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces.
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The orientation on a real vector space is the arbitrary choice of which ordered bases are " positively " oriented and which are " negatively " oriented.
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More generally, any real vector space of ( finite ) dimension " D " can be viewed as the multiplication by a real number.
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