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 $\struct {\map C I, +, \, \cdot \,}_\R$ be the vector space of real-valued functions, continuous on $I$.
Let $\map x t \in \map C I$ 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 $\struct {\map C I, +, \, \cdot \,}_\R$.
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 Function is Completely Multiplicative | |||||||||||
\(\ds \) | \(=\) | \(\ds \size \alpha \max_{t \mathop \in I} \size {\map x t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \size \alpha \sup_{t \mathop \in I} \size {\map x t}\) | 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
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed spaces