# Definition:Real Vector Space

## Definition

Let $\R$ be the set of real numbers.

Then the $\R$-module $\R^n$ is called the **real ($n$-dimensional) vector space**.

## Also known as

A **real vector space** is also known as a **real linear space**.

Some sources refer to this as a **real Euclidean space**, based on the fact that $\R^2$ and $\R^3$ support Euclidean geometry.

However, this latter term is used on $\mathsf{Pr} \infty \mathsf{fWiki}$ in the context of metric spaces to define a real Cartesian space which has the Euclidean metric applied to it.

The constructs are in fact the same thing, but the emphasis is different.

## Also see

This object is proved to be a vector space in Real Vector Space is Vector Space.

The definition is also expanded upon in Real Numbers form Vector Space.

The real vector spaces have direct applications to the real world. In fact, it could be suggested that they are the interface between mathematics and physical reality, as follows:

- From the definition of the real number line, the $\R$-vector space $\R$ is isomorphic to $\R$ to an infinite straight line.

- From the definition of the real number plane, the $\R$-vector space $\R^2$ is isomorphic to $\R$ to an infinite flat plane.

- The $\R$-vector space $\R^3$ can be shown (given appropriate assumptions about the nature of the universe) to be isomorphic to the spatial universe.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.2$ - 1994: Martin J. Osborne and Ariel Rubinstein:
*A Course in Game Theory*... (previous) ... (next): $1.7$: Terminology and Notation - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.2$