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

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

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

We have that:

Real Numbers form Ordered Field
Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements

Therefore, $M$ has the greatest element.

### Norm Axiom $\text N 1$: Positive Definiteness

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

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

Suppose $\norm M_\infty = 0$.

Then:

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

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

$\Box$

### Norm Axiom $\text N 2$: Positive Homogeneity

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

$\Box$

### Norm Axiom $\text N 3$: Triangle Inequality

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

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

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

We have that:

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

This holds for any $i, j$.

Hence:

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

$\Box$

All norm axioms are seen to be satisfied.

Hence the result.

$\blacksquare$