To quote an example with real vector spaces, just think to the Banach space L ^ 1 ( [ 0, 1 ] ) with the subspace of the continuous functions, and of the simple functions.
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The idea is to abstract from the regulator of a number field to some " higher regulator " ( the Beilinson regulator ), a determinant constructed on a real vector space that comes from algebraic K-theory.
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More generally, given a converging series of vectors in a finite-dimensional real vector space " E ", the set of sums of converging rearranged series is an affine subspace of " E ".
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This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves, realizing it as a group of rotations among them, but it also acts on the column vectors ( that is, the spinors ).
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More formally, given a finite number of points x _ 1, x _ 2, \ dots, x _ n \, in a real vector space, a convex combination of these points is a point of the form
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A "'weakly Luzin set "'is an uncountable subset of a real vector space such that for any uncountable subset the set of directions between different elements of the subset is dense in the sphere of directions.
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More precisely, I'll assume we have a real affine plane " P ", that is, a set on which a two-dimensional real vector space " V " acts simply and transitively.
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In mathematics, a "'quasiconvex function "'is a convex subset of a real vector space such that the inverse image of any set of the form (-\ infty, a ) is a convex set.
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There are several variations, where 2 " ? " is replaced by the real numbers or a real vector space or the unit interval, and the number 1 / 2 is replaced by some positive number " ? ".
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The main example is the case that the real vector space " V " is a complex structure " J " that is an orthogonal transformation with respect to the inner product " g " on " V ".
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