Without reference to coordinates, this is the space of lines through the origin in an ( " n " + 1 )-dimensional real vector space.
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In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice dimensions, where multiplication to the imaginary unit.
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A two-dimensional real vector space with the Minkowski inner product is called-dimensional Minkowski space, often denoted "'R "'1, 1.
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As a concrete example, every even-dimensional real vector space is isomorphic to the underlying real space of a complex vector space : it admits a linear complex structure.
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Consider an infinite dimensional real vector space X, and the topology generated by the finite codimensional affine subspaces of X ( that is, open sets are union of these ).
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:Also note that there is no canonical isomorphism between V and V * if V is a plain real vector space of finite dimension > 1, with no additional structure.
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For every subset " Q " of a real vector space, its is the interval of real numbers [ 0, 1 ], which contains the integer end-points.
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The complex numbers, real vector spaces, and indeed any metric space then naturally inherit the apartness relation of the real numbers, even though they do not come equipped with any natural ordering.
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All such complex structures can be obtained as follows : take a lattice ? in "'C " "'n " considered as real vector space; then the quotient group
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Irreducible representations over the reals in the case when " V " is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details.
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