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Concept# Real coordinate space

Summary

In mathematics, the real coordinate space of dimension n, denoted Rn or , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers.
Special cases are called the real line R1 and the real coordinate plane R2.
With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors.
The coordinates over any basis of the elements of a real vector space form a real coordinate space of the same dimension as that of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension n form a real coordinate space of dimension n.
These one to one correspondences between vectors, points and coordinate vectors explain the names of coordinate space and coordinate vector. It allows using geometric terms and methods for studying real coordinate spaces, and, conversely, to use methods of calculus in geometry. This approach of geometry was introduced by René Descartes in the 17th century. It is widely used, as it allows locating points in Euclidean spaces, and computing with them.
For any natural number n, the set Rn consists of all n-tuples of real numbers (R). It is called the "n-dimensional real space" or the "real n-space".
An element of Rn is thus a n-tuple, and is written
where each xi is a real number. So, in multivariable calculus, the domain of a function of several real variables and the codomain of a real vector valued function are subsets of Rn for some n.
The real n-space has several further properties, notably:
With componentwise addition and scalar multiplication, it is a real vector space. Every n-dimensional real vector space is isomorphic to it.
With the dot product (sum of the term by term product of the components), it is an inner product space. Every n-dimensional real inner product space is isomorphic to it.
As every inner product space, it is a topological space, and a topological vector space.
It is a Euclidean space and a real affine space, and every Euclidean or affine space is isomorphic to it.

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Pseudo-Euclidean space

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, ..., en), be applied to a vector x = x1e1 + ⋯ + xnen, giving which is called the scalar square of the vector x. For Euclidean spaces, k = n, implying that the quadratic form is positive-definite. When 0 < k < n, q is an isotropic quadratic form, otherwise it is anisotropic.

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