Real Numbers form Algebra
Contents
Theorem
The set of real numbers $\R$ forms an algebra over the field of real numbers.
This algebra is:
- $(1): \quad$ An associative algebra.
- $(2): \quad$ A commutative algebra.
- $(3): \quad$ A normed division algebra.
- $(4): \quad$ A nicely normed $*$-algebra whose $*$ operator is the identity mapping.
- $(5): \quad$ A real $*$-algebra.
Proof
Construction of Algebra
From Real Numbers form Field, $\left({\R, +, \times}\right)$ is a field.
Let this be expressed as $\left({\R, +_\R, \times_\R}\right)$ in order to call attention to the precise scope of the operators.
From Real Numbers form Vector Space, we have that $\left({\R^1, +, \cdot}\right)_\R$ is a vector space, where:
- the field is $\left({\R, +_\R, \times_\R}\right)$
- the abelian group is $\left({\R, +_G}\right)$ where $+_G$ is real addition.
In Real Numbers form Vector Space, it is established that elements of $\left({\R^1, +, \cdot}\right)_\R$ are in fact just real numbers.
So, let $\times$ be the binary operation on $\left({\R^1, +, \cdot}\right)_\R$ defined as:
- $\forall x, y \in \left({\R^1, +, \cdot}\right)_\R: x \times y = x \times_\R y$
where $\times_\R$ is real multiplication.
Proof of an Algebra
We need to show that $\times$ as defined on $\left({\R^1, +, \cdot}\right)_\R$ as:
- $\forall x, y \in \left({\R^1, +, \cdot}\right)_\R = x \times_\R y$
is bilinear.
That is: $\forall a, b \in \R, x, y \in \R^1$:
- $\left({\left({a \cdot x}\right) + \left({b \cdot y}\right)}\right) \times z = \left({a \cdot \left({x \times z}\right)}\right) + \left({b \cdot \left({y \times z}\right)}\right)$
- $z \times \left({\left({a \cdot x}\right) + \left({b \cdot y}\right)}\right) = \left({a \cdot \left({z \times x}\right)}\right) + \left({b \cdot \left({z \times y}\right)}\right)$
So:
| \(\displaystyle \left({\left({a \cdot x}\right) + \left({b \cdot y}\right)}\right) \times z\) | \(=\) | \(\displaystyle \left({\left({a \times_\R x}\right) + \left({b \times_\R y}\right)}\right) \times_\R z\) | $\cdot$ and $\times$ are Real Multiplication | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({\left({a \times_\R x}\right) \times_\R z}\right) + \left({\left({b \times_\R y}\right) \times_\R z}\right)\) | Real Multiplication Distributes over Addition | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({a \times_\R \left({x \times_\R z}\right)}\right) + \left({b \times_\R \left({y \times_\R z}\right)}\right)\) | Real Multiplication is Associative | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({a \cdot \left({x \times z}\right)}\right) + \left({b \cdot \left({y \times z}\right)}\right)\) | $\cdot$ and $\times$ are Real Multiplication |
Similarly:
| \(\displaystyle z \times \left({\left({a \cdot x}\right) + \left({b \cdot y}\right)}\right)\) | \(=\) | \(\displaystyle z \times_\R \left({\left({a \times_\R x}\right) + \left({b \times_\R y}\right)}\right)\) | $\cdot$ and $\times$ are Real Multiplication | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({z \times_\R \left({a \times_\R x}\right)}\right) + \left({z \times_\R \left({b \times_\R y}\right)}\right)\) | Real Multiplication Distributes over Addition | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({\left({z \times_\R a}\right) \times_\R x}\right) + \left({\left({z \times_\R b}\right) \times_\R y}\right)\) | Real Multiplication is Associative | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({\left({a \times_\R z}\right) \times_\R x}\right) + \left({\left({b \times_\R z}\right) \times_\R y}\right)\) | Real Multiplication is Commutative | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({a \times_\R \left({z \times_\R x}\right)}\right) + \left({b \times_\R \left({z \times_\R y}\right)}\right)\) | Real Multiplication is Associative | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({a \cdot \left({z \times x}\right)}\right) + \left({b \cdot \left({z \times y}\right)}\right)\) | $\cdot$ and $\times$ are Real Multiplication |
So the set of real numbers forms an algebra $\left({\R, \times}\right)$.
$\Box$
Proof of Associativity
Elements of $\left({\R, \times}\right)$ are merely real numbers, and $\times$ is just real multiplication.
Associativity of $\times$ therefore follows directly from Real Multiplication is Associative.
$\Box$
Proof of Commutativity
Elements of $\left({\R, \times}\right)$ are merely real numbers, and $\times$ is just real multiplication.
Associativity of $\times$ therefore follows directly from Real Multiplication is Commutative.
$\Box$
Proof of Normed Division Algebra
Elements of $\left({\R, \times}\right)$ are merely real numbers, and $\times$ is just real multiplication.
So from Real Multiplication Identity is One, $\left({\R, \times}\right)$ has a unit, which is $1$.
So $\left({\R, \times}\right)$ is a unitary algebra.
From Inverses for Real Multiplication, every element of $\left({\R, \times}\right)$ except $0$ has a multiplicative inverse.
So $\left({\R, \times}\right)$ is a division algebra and hence a unitary division algebra.
We define a norm on $\left({\R, \times}\right)$ by:
- $\forall a \in \R: \left \Vert {a} \right \Vert = \left \vert {a} \right \vert = \sqrt {a^2}$
That is, by the absolute value of $a$.
This is a norm because:
- $(1): \quad \left \Vert x \right \Vert = 0 \iff x = \mathbf 0$
- $(2): \quad \left \Vert \lambda x \right \Vert = \left \vert \lambda \right \vert \left \vert x \right \vert = \left \vert \lambda \right \vert \left \Vert x \right \Vert$
- $(3): \quad \left \Vert x - y \right \Vert \le \left \Vert x - z \right \Vert + \left \Vert z - y \right \Vert$ which follows from Real Number Line is Metric Space.
It also follows that:
- $\left \Vert x \times y \right \Vert = \left \vert x \times y \right \vert = \left \vert x \right \vert \times \left \vert y \right \vert = \left \Vert x \right \Vert \times \left \Vert y \right \Vert$
and so $\left({\R, \times}\right)$ is a normed division algebra.
$\Box$
Proof of Nicely Normed $*$-Algebra
We define the conjugation $*$ by making it the identity mapping on $\R$.
That is:
- $\forall a \in \R: a^* = a$
We have that:
| \((1):\quad\) | \(\displaystyle \left({a^*}\right)^*\) | \(=\) | \(\displaystyle a^*\) | Definition of $*$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle a\) | Definition of $*$ |
| \((2):\quad\) | \(\displaystyle \left({a + b}\right)^*\) | \(=\) | \(\displaystyle a + b\) | Definition of $*$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle b + a\) | Real Addition is Commutative | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle b^* + a^*\) |
demonstrating that $*$ is indeed a conjugation.
Then we have that:
| \((3):\quad\) | \(\displaystyle a + a^*\) | \(=\) | \(\displaystyle a + a\) | Definition of $*$ | |||||||||
| \(\displaystyle \) | \(\in\) | \(\displaystyle \R\) |
| \((4):\quad\) | \(\displaystyle a \times a^*\) | \(=\) | \(\displaystyle a \times a\) | Definition of $*$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle a^2\) | |||||||||||
| \(\displaystyle \) | \(>\) | \(\displaystyle 0\) | Square of Real Number is Non-Negative |
Similarly for $a^* + a$.
Trivially, $a^* + a$ and $a \times a^*$ are both real.
So $\left({\R, \times}\right)$ is a nicely normed $*$-algebra.
$\Box$
Proof of Real $*$-Algebra
By definition of $*$:
- $\forall a \in \R: a^* = a$
Hence, trivially:
- $\forall a \in \R: a^* \in \R$
That is, $\left({\R, \times}\right)$ is a real $*$-algebra.
$\blacksquare$
