=== 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 Abelian 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: :$\displaystyle \R^n = \prod_{i \mathop = 1}^n \struct {\R, +_G} = \underbrace {\struct {\R, +_G} \times \cdots \times \struct {\R, +_G} }_{n \text{ copies} }$ 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: $(1): \quad \lambda \cdot \paren {\mathbf x + \mathbf y} = \paren {\lambda \cdot \mathbf x} + \paren {\lambda \cdot \mathbf y}$: {{begin-eqn}} {{eqn | l = \lambda \cdot \paren {\mathbf x + \mathbf y}       | r = \tuple {\lambda \times_\R \paren {x_1 +_G y_1}, \lambda \times_\R \paren {x_2 +_G y_2}, \ldots, \lambda \times_\R \paren {x_n +_G y_n} }       | c = Definition of $\cdot$ over $\R \times \R^n$ }} {{eqn | r = \tuple {\paren {\lambda \times_\R x_1 +_G \lambda \times_\R y_1}, \paren {\lambda \times_\R x_2 +_G \lambda \times_\R y_2}, \ldots, \paren {\lambda \times_\R x_n +_G \lambda \times_\R y_n} }       | c = $\times_\R$ distributes over $+_G$ }} {{eqn | r = \tuple {\lambda \times_\R x_1, \lambda \times_\R x_2, \ldots, \lambda \times_\R x_n} + \tuple {\lambda \times_\R y_1, \lambda \times_\R y_2, \ldots, \lambda \times_\R y_n}       | c = Definition of $+$ over $\R^n$ }} {{eqn | r = \lambda \cdot \tuple {x_1, x_2, \ldots, x_n} + \lambda \cdot \tuple {y_1, y_2, \ldots, y_n}       | c = Definition of $\cdot$ over $\R \times \R^n$ }} {{eqn | r = \lambda \cdot \mathbf x + \lambda \cdot \mathbf y       | c = Definition of $\mathbf x$ and $\mathbf y$ }} {{end-eqn}} So $(1)$ has been shown to hold. $(2): \quad \paren {\lambda +_\R \mu} \cdot x = \paren {\lambda \cdot \mathbf x} + \paren {\mu \cdot \mathbf x}$: {{begin-eqn}} {{eqn | l = \paren {\lambda +_\R \mu} \cdot x       | r = \tuple {\paren {\lambda +_\R \mu} \times_\R x_1, \paren {\lambda +_\R \mu} \times_\R x_2, \ldots, \paren {\lambda +_\R \mu} \times_\R x_n}       | c = Definition of $\cdot$ over $\R \times \R^n$ }} {{eqn | r = \tuple {\paren {\lambda \times_\R x_1 +_G \mu \times_\R x_1}, \paren {\lambda \times_\R x_2 +_G \mu \times_\R x_2}, \ldots, \paren {\lambda \times_\R x_n +_G \mu \times_\R x_n} }       | c = $\times_\R$ distributes over $+_\R$, and $+_\R$ is the same operation as $+_G$ }} {{eqn | r = \tuple {\lambda \times_\R x_1, \lambda \times_\R x_2, \ldots, \lambda \times_\R x_n} + \tuple {\mu \times_\R x_1, \mu \times_\R x_2, \ldots, \mu \times_\R x_n}       | c = Definition of $+$ over $\R^n$ }} {{eqn | r = \lambda \cdot \tuple {x_1, x_2, \ldots, x_n} + \mu \cdot \tuple {x_1, x_2, \ldots, x_n}       | c = Definition of $\cdot$ over $\R \times \R^n$ }} {{eqn | r = \lambda \cdot \mathbf x + \mu \cdot \mathbf x       | c = Definition of $\mathbf x$ and $\mathbf y$ }} {{end-eqn}} So $(2)$ has been shown to hold. $(3): \quad \paren {\lambda \times_\R \mu} \cdot x = \lambda \cdot \paren {\mu \cdot \mathbf x}$: {{begin-eqn}} {{eqn | l = \paren {\lambda \times_\R \mu} \cdot x       | r = \tuple {\paren {\lambda \times_\R \mu} \times_\R x_1, \paren {\lambda \times_\R \mu} \times_\R x_2, \ldots, \paren {\lambda \times_\R \mu} \times_\R x_n}       | c = Definition of $\cdot$ over $\R \times \R^n$ }} {{eqn | r = \tuple {\lambda \times_\R \paren {\mu \times_\R x_1}, \lambda \times_\R \paren {\mu \times_\R x_2}, \ldots, \lambda \times_\R \paren {\mu \times_\R x_n} }       | c = Real Multiplication is Associative }} {{eqn | r = \lambda \cdot \tuple {\mu \times_\R x_1, \mu \times_\R x_2, \ldots, \mu \times_\R x_n}       | c = Definition of $\cdot$ over $\R \times \R^n$ }} {{eqn | r = \lambda \cdot \paren {\mu \cdot \tuple {x_1, x_2, \ldots, x_n} }       | c = Definition of $\cdot$ over $\R \times \R^n$ }} {{eqn | r = \lambda \cdot \paren {\mu \cdot \mathbf x}       | c = Definition of $\mathbf x$ }} {{end-eqn}} So $(3)$ has been shown to hold. $(4): \quad \forall \mathbf x \in \R^n: 1 \cdot \mathbf x = \mathbf x$: {{begin-eqn}} {{eqn | l = 1 \cdot \mathbf x = \mathbf x       | r = \tuple {1 \times_\R x_1, 1 \times_\R x_2, \ldots, 1 \times_\R x_n}       | c = Definition of $\cdot$ over $\R \times \R^n$ }} {{eqn | r = \tuple {x_1, x_2, \ldots, x_n}       | c = Real Multiplication Identity is One }} {{eqn | r = \mathbf x       | c = Definition of $\mathbf x$ }} {{end-eqn}} So $(4)$ has been shown to hold. So the $\R$-module $\R^n$ is a vector space, as we were to prove. {{qed}}