In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted Rn or R n {\displaystyle \mathbb {R} ^{n}} , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors. Special cases are called the real line R1, the real coordinate plane R2, and the real coordinate three-dimensional space R3. With component-wise addition and scalar multiplication, it is a real vector space. 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, En (Euclidean line, E; Euclidean plane, E2; Euclidean three-dimensional space, E3) 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.
Definition and structures 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
( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})}
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. It is an analytic manifold, and can be considered as the prototype of all manifolds, as, by definition, a manifold is, near each point, isomorphic to an open subset of Rn. It is an algebraic variety, and every real algebraic variety is a subset of Rn. These properties and structures of Rn make it fundamental in almost all areas of mathematics and their application domains, such as statistics, probability theory, and many parts of physics.
The domain of a function of several variables
Any function f(x1, x2, ..., xn) of n real variables can be considered as a function on Rn (that is, with Rn as its domain). The use of the real n-space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions. Consider, for n = 2, a function composition of the following form:
F ( t ) = f ( g 1 ( t ) , g 2 ( t ) ) , {\displaystyle F(t)=f(g_{1}(t),g_{2}(t)),}
where functions g1 and g2 are continuous. If
∀x1 ∈ R : f(x1, ·) is continuous (by x2) ∀x2 ∈ R : f(·, x2) is continuous (by x1) then F is not necessarily continuous. Continuity is a stronger condition: the continuity of f in the natural R2 topology (discussed below), also called multivariable continuity, which is sufficient for continuity of the composition F.
Vector space The coordinate space Rn forms an n-dimensional vector space over the field of real numbers with the addition of the structure of linearity, and is often still denoted Rn. The operations on Rn as a vector space are typically defined by
x + y = ( x 1 + y 1 , x 2 + y 2 , … , x n + y n ) {\displaystyle \mathbf {x} +\mathbf {y} =(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n})}
α x = ( α x 1 , α x 2 , … , α x n ) . {\displaystyle \alpha \mathbf {x} =(\alpha x_{1},\alpha x_{2},\ldots ,\alpha x_{n}).}
The zero vector is given by
0 = ( 0 , 0 , … , 0 ) {\displaystyle \mathbf {0} =(0,0,\ldots ,0)}
and the additive inverse of the vector x is given by
− x = ( − x 1 , − x 2 , … , − x n ) . {\displaystyle -\mathbf {x} =(-x_{1},-x_{2},\ldots ,-x_{n}).}
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