# Supremum Norm on Vector Space of Real Matrices is Norm

## Theorem

Supremum Norm forms a norm on the vector space of real matrices.

## Proof

Let $M \in \R^{m \times n} : m, n \in \N_{\mathop > 0}$ be a real matrix.

Denote the $\paren {i, j}$-th entry of $M$ by $a_{ij}$.

Note that the set of matrix elements of $M$ is a finite set of real numbers.

We have that:

Real Numbers form Ordered Field
Non-Empty Finite Set has Greatest Element

Therefore, $M$ has the greatest element.

### Norm Axiom $(\text N 1)$

 $\displaystyle \norm M_\infty$ $=$ $\displaystyle \max_{\begin {split} 1 \mathop \le i \mathop \le m \\ 1 \mathop \le j \mathop \le n \end {split} } \size {a_{ij} }$ Greatest Element is Supremum, Definition of Max Operation $\displaystyle$ $\ge$ $\displaystyle 0$

Equality is obtained for $M$ being a zero matrix.

Suppose $\norm M_\infty = 0$.

Then:

$\displaystyle \forall i, j : 1 \le i \le m, 1 \le j \le n : \size {a_{ij}} = 0$

In other words, $M$ is a zero matrix.

$\Box$

### Norm Axiom $(\text N 2)$

 $\displaystyle \norm {\alpha \cdot M}_\infty$ $=$ $\displaystyle \max_{\begin {split} & 1 \mathop \le i \mathop \le m\\ & 1 \mathop \le j \mathop \le n \end {split} } \size {\alpha m_{ij} }$ Greatest Element is Supremum $\displaystyle$ $=$ $\displaystyle \max_{\begin {split} & 1 \mathop \le i \mathop \le m\\ & 1 \mathop \le j \mathop \le n \end {split} } \size \alpha \size {a_{ij} }$ Absolute Value of Product $\displaystyle$ $=$ $\displaystyle \size \alpha \max_{\begin {split} & 1 \mathop \le i \mathop \le m\\ & 1 \mathop \le j \mathop \le n \end {split} } \size {a_{ij} }$ $\displaystyle$ $=$ $\displaystyle \size \alpha \norm M_\infty$ Greatest Element is Supremum

$\Box$

### Norm Axiom $(\text N 3)$

Let $P, Q \in \R^{m \times n}$.

Denote their $\paren {i, j}$-th matrix elements as $p_{ij}$ and $q_{ij}$ respectively.

Fix $i,j \in \N : 1 \le i \le m, 1 \le j \le n$.

We have that:

 $\displaystyle \size {p_{ij} + q_{ij} }$ $\le$ $\displaystyle \size {p_{ij} } + \size {q_{ij} }$ Triangle Inequality for Real Numbers $\displaystyle$ $\le$ $\displaystyle \max_{\begin {split} & 1 \mathop \le i \mathop \le m\\ & 1 \mathop \le j \mathop \le n \end {split} } \size {p_{ij} } + \max_{\begin {split} & 1 \mathop \le i \mathop \le m\\ & 1 \mathop \le j \mathop \le n \end {split} } \size {q_{ij} }$ Definition of Max Operation $\displaystyle$ $=$ $\displaystyle \norm P_\infty + \norm Q_\infty$ Greatest Element is Supremum, Definition of Supremum Norm

This holds for any $i,j$.

Hence:

 $\displaystyle \norm {P + Q}_\infty$ $=$ $\displaystyle \max_{\begin {split} & 1 \mathop \le i \mathop \le m\\ & 1 \mathop \le j \mathop \le n \end {split} } \size {p_{ij} + q_{ij} }$ Greatest Element is Supremum $\displaystyle$ $\le$ $\displaystyle \norm P_\infty + \norm Q_\infty$

$\Box$

All norm axioms are seen to be satisfied.

Hence the result.

$\blacksquare$