Definition:Real Vector Space

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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:

  • 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.

  • Results about real vector spaces can be found here.