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

$\blacksquare$


Sources