some structures on n

Let n{1,2,}. Then, as a set, n is the n-fold Cartesian product of the real numbers.

0.0.1 Vector space structure of n

If u=(u1,,un) and v=(v1,,vn) are points in n, we define their sum as


Also, if λ is a scalar (real number), then scalar multiplication is defined as


With these operationsMathworldPlanetmath, n becomes a vector spaceMathworldPlanetmath (over ) with dimensionMathworldPlanetmathPlanetmath n. In other words, with this structureMathworldPlanetmath, we can talk about, vectors, lines, subspacesMathworldPlanetmathPlanetmathPlanetmath of different dimension.

0.0.2 Inner product for n

For u and v as above, we define the inner product as


With this productPlanetmathPlanetmathPlanetmath, n is called an Euclidean spaceMathworldPlanetmath.

We have also an induced norm u=u,u, which gives n the structure of a normed space (and thus metric space). This inner product let us talk about length, angle between vectors, orthogonal vectorsMathworldPlanetmath.

0.0.3 Topology for n

The usual topology for n is the topologyMathworldPlanetmathPlanetmath induced by the metric


As a basis for the topology induced by the above norm, one can take open balls B(x,r)={ynx-y<r} where r>0 and xn.

Properties of the topological space n are:

  1. 1.

    n is second countable, i.e., n has a countable basis.

  2. 2.

    (Heine-Borel theorem) A set in n is compactPlanetmathPlanetmath if and only if it is closed and boundedPlanetmathPlanetmathPlanetmath.

  3. 3.

    Since n is a metric space, n is a Hausdorff space.

Title some structures on n
Canonical name SomeStructuresOnmathbbRn
Date of creation 2013-03-22 14:03:47
Last modified on 2013-03-22 14:03:47
Owner drini (3)
Last modified by drini (3)
Numerical id 10
Author drini (3)
Entry type Definition
Classification msc 54E35
Classification msc 53A99