Real Vector Space is Vector Space

== Theorem ==
Let $\R$ be the set of real numbers.
Then the real vector space $\R^n$ is a vector space.
== Proof ==
=== Construction of Real Vector Space ===
From the definition, a vector space is a unitary module whose scalar ring is a field.
In order to call attention to the precise scope of the operators, let real addition and real multiplication be expressed as $+_\R$ and $\times_\R$ respectively.
Then we can express the field of real numbers as $\struct {\R, +_\R, \times_\R}$.
From Real Numbers under Addition form Group, $\struct {\R, +}$ is a group.
Again, in order to call attention to the precise scope of the operator, let real addition be expressed on $\struct {\R, +}$ as $+_G$.
That is, the group under consideration is $\struct {\R, +_G}$.
Consider the cartesian product:
:$\ds \R^n = \prod_{i \mathop = 1}^n \struct {\R, +_G} = \underbrace {\struct {\R, +_G} \times \cdots \times \struct {\R, +_G} }_{n \text{ copies} }$
Let:
:$\mathbf a = \tuple {a_1, a_2, \ldots, a_n}$
:$\mathbf b = \tuple {b_1, b_2, \ldots, b_n}$
be arbitrary elements of $\R^n$.
Let $\lambda$ be an arbitrary element of $\R$.
Let $+$ be the binary operation defined on $\R^n$ as:
:$\mathbf a + \mathbf b = \tuple {a_1 +_G b_1, a_2 +_G b_2, \ldots, a_n +_G b_n}$
Also let $\cdot$ be the binary operation defined on $\R \times \R^n$ as:
:$\lambda \cdot \mathbf a = \tuple {\lambda \times_\R a_1, \lambda \times_\R a_2, \ldots, \lambda \times_\R a_n}$
In this context, $\lambda \times_\R a_j$ is defined as real multiplication, as is appropriate (both $\lambda$ and $a_j$ are real numbers).
With this set of definitions, the structure $\struct {\R^n, +, \cdot}$ is a vector space, as is shown in Proof of Real Vector Space below.
=== Proof of Real Vector Space ===
In order to show that $\struct {\R^n, +, \cdot}$ is a vector space, we need to show that:
$\forall \mathbf x, \mathbf y \in \R^n, \forall \lambda, \mu \in \R$:
:$(1): \quad \lambda \cdot \paren {\mathbf x + \mathbf y} = \paren {\lambda \cdot \mathbf x} + \paren {\lambda \cdot \mathbf y}$
:$(2): \quad \paren {\lambda +_\R \mu} \cdot x = \paren {\lambda \cdot \mathbf x} + \paren {\mu \cdot \mathbf x}$
:$(3): \quad \paren {\lambda \times_\R \mu} \cdot x = \lambda \cdot \paren {\mu \cdot \mathbf x}$
:$(4): \quad \forall \mathbf x \in \R^n: 1 \cdot \mathbf x = \mathbf x$.
where $1$ in this context means the real number one.
From External Direct Product of Groups is Group, we have that $\struct {\R^n, +}$ is a group in its own right.
Let:
:$\mathbf x = \tuple {x_1, x_2, \ldots, x_n}$
:$\mathbf y = \tuple {y_1, y_2, \ldots, y_n}$
Checking the criteria in turn:
$(1): \quad \lambda \cdot \paren {\mathbf x + \mathbf y} = \paren {\lambda \cdot \mathbf x} + \paren {\lambda \cdot \mathbf y}$:
So $(1)$ has been shown to hold.
$(2): \quad \paren {\lambda +_\R \mu} \cdot x = \paren {\lambda \cdot \mathbf x} + \paren {\mu \cdot \mathbf x}$:
So $(2)$ has been shown to hold.
$(3): \quad \paren {\lambda \times_\R \mu} \cdot x = \lambda \cdot \paren {\mu \cdot \mathbf x}$:
So $(3)$ has been shown to hold.
$(4): \quad \forall \mathbf x \in \R^n: 1 \cdot \mathbf x = \mathbf x$:
So $(4)$ has been shown to hold.
So the $\R$-module $\R^n$ is a vector space, as we were to prove.
== Warning ==
Notice that the factors of $\R^n$ are considered to be elements of a group, where $+_\R$ is the only operator defined.
That is, in this particular context, $\times_G$ is not defined and can not be used.
This stands to reason, as in the context of a vector space, there is no unique and canonical definition for multiplication of vectors.
Several are defined (for example, dot product and cross product), but these are not canonical.
== Also see ==
It follows directly, by setting $n = 1$, that the $\R$-module $\R$ itself can also be regarded as a vector space.
This is expanded upon in Real Numbers form Vector Space.
== Sources ==
* : Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules
* : Chapter $1.1$: Normed and Banach spaces. Vector Spaces
Category:Examples of Vector Spaces
Category:Real Numbers
Category:Analytic Geometry