# Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function

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## Theorem

Let $I = \closedint a b$ be a closed interval.

Let $\struct {\map \CC I, +, \, \cdot \,}_\R$ be the vector space of real-valued functions, continuous on $I$.

Let $\map x t \in \map \CC I$ be a continuous real function.

Let $\size {\, \cdot \,}$ be the absolute value.

Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on real-valued functions, continuous on $I$.

Then $\norm {\, \cdot \,}_\infty$ is a norm over $\struct {\map \CC I, +, \, \cdot \,}_\R$.

## Proof

### Positive definiteness

 $\ds \norm x_\infty$ $=$ $\ds \sup_{t \mathop \in I} \size {\map x t}$ Definition of Supremum Norm on Space of Continuous on Closed Interval Real-Valued Functions $\ds$ $=$ $\ds \max_{t \mathop \in I} \size {\map x t}$ Weierstrass Extreme Value Theorem $\ds$ $\ge$ $\ds \size {\map x t}$ Definition of Max Operation $\ds$ $\ge$ $\ds 0$ Complex Modulus is Non-Negative

Suppose $\norm x_\infty = 0$.

Then:

 $\ds 0$ $=$ $\ds \norm x_\infty$ $\ds$ $=$ $\ds \sup_{t \mathop \in I} \size {\map x t}$ Definition of Supremum Norm on Space of Continuous on Closed Interval Real-Valued Functions $\ds$ $=$ $\ds \max_{t \mathop \in I} \size {\map x t}$ Weierstrass Extreme Value Theorem $\ds$ $\ge$ $\ds \size {\map x t}$ Definition of Max Operation $\ds$ $\ge$ $\ds 0$ Complex Modulus is Non-Negative $\ds \leadsto \ \$ $\ds \size {\map x t}$ $=$ $\ds 0$ $\ds \leadsto \ \$ $\ds \map x t$ $=$ $\ds 0$ Complex Modulus equals Zero iff Zero

Therefore:

$\forall t \in I : \map x t = 0$

$\Box$

### Positive homogeneity

Let $\alpha \in \R$.

 $\ds \norm {\alpha \cdot x}_\infty$ $=$ $\ds \sup_{t \mathop \in I} \size {\map {\paren {\alpha \cdot x} } t}$ Definition of Supremum Norm on Space of Continuous on Closed Interval Real-Valued Functions $\ds$ $=$ $\ds \max_{t \mathop \in I} \size {\map {\paren {\alpha \cdot x} } t}$ Weierstrass Extreme Value Theorem $\ds$ $=$ $\ds \max_{t \mathop \in I} \size {\alpha {\map x t} }$ Definition of Pointwise Scalar Multiplication of Real-Valued Functions $\ds$ $=$ $\ds \max_{t \mathop \in I} \size \alpha \size {\map x t}$ Absolute Value of Product $\ds$ $=$ $\ds \size \alpha \max_{t \mathop \in I} \size {\map x t}$ $\ds$ $=$ $\ds \size \alpha \sup_{t \mathop \in I} \size {\map x t}$ Weierstrass Extreme Value Theorem $\ds$ $=$ $\ds \size \alpha \norm x_\infty$ Definition of Supremum Norm on Space of Continuous on Closed Interval Real-Valued Functions

$\Box$

### Triangle inequality

 $\ds \size {\map {\paren {x_1 + x_2} } t}$ $=$ $\ds \size {\map {x_1} t + \map {x_2} t}$ Definition of Pointwise Addition of Real-Valued Functions $\ds$ $\le$ $\ds \size {\map {x_1} t} + \size {\map {x_2} t}$ Triangle Inequality for Real Numbers $\ds$ $\le$ $\ds \max_{t \mathop \in I} \size {\map {x_1} t} + \max_{t \mathop \in I} \size {\map {x_2} t}$ Definition of Max Operation $\ds$ $=$ $\ds \sup_{t \mathop \in I} \size {\map {x_1} t } + \sup_{t \mathop \in I} \size { \map {x_2} t }$ Weierstrass Extreme Value Theorem $\ds$ $=$ $\ds \norm {x_1}_\infty + \norm {x_2}_\infty$ Definition of Supremum Norm on Space of Continuous on Closed Interval Real-Valued Functions

$\blacksquare$