C^k Norm is Norm

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Theorem

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

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

Let $x \in \map {\CC^k} I$ be a real-valued function of differentiability class $k$.

Let $\norm {\, \cdot \,}_{\map {C^k} I}$ be the $C^k$ norm on $I$.


Then $\norm {\, \cdot \,}_{\map {C^k} I}$ is a norm on $\struct {\map {\CC^k} I, +, \, \cdot \,}_\R$.

Proof

Positive definiteness

Let $x \in \map {\CC^k} I$.

Then:

\(\displaystyle \norm x_{\map {C^k} I}\) \(=\) \(\displaystyle \sum_{i \mathop = 0}^k \norm {x^{\paren i} }_\infty\)
\(\displaystyle \) \(\ge\) \(\displaystyle \sum_{i \mathop = 0}^k 0\) Supremum Norm is Norm, Norm Axiom $\paren {N1}:$ Positive Definiteness
\(\displaystyle \) \(=\) \(\displaystyle 0\)

Suppose $\norm x_{\map {C^k} I} = 0$.

We have that the sum of non-negatives is zero if every element is zero.

Hence:

$\forall i \in \N : 0 \le i \le k : \norm {x^{\paren i}}_\infty = 0$

Namely, $\norm x_\infty = 0$.

By Supremum Norm is Norm and Norm Axiom $\paren {N1}:$ Positive Definiteness:

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


Positive homogeneity

Let $x \in \map {\CC^k} I$, $\alpha \in \R$.

Then:

\(\displaystyle \norm {\alpha x}_{\map {C^k} I}\) \(=\) \(\displaystyle \sum_{i \mathop = 0}^k \norm {\paren {\alpha x}^{\paren i} }_\infty\) Definition of C^k Norm
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 0}^k \norm {\alpha x^{\paren i} }_\infty\) Definition of Pointwise Scalar Multiplication of Real-Valued Functions
\(\displaystyle \) \(=\) \(\displaystyle \size \alpha \sum_{i \mathop = 0}^k \norm {x^{\paren i} }_\infty\) Supremum norm on continuous real-valued functions is a norm: positive homogeneity
\(\displaystyle \) \(=\) \(\displaystyle \size \alpha \norm {x}_{\map {C^k} I}\) Definition of C^k Norm


Triangle inequality

Let $x, y \in \map {\CC^k} I$

\(\displaystyle \norm {x + y}_{\map {C^k} I}\) \(=\) \(\displaystyle \sum_{i \mathop = 0}^k \norm {\paren {x + y}^{\paren i} }_\infty\) Definition of Pointwise Addition of Real-Valued Functions
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 0}^k \norm {x^{\paren i} + y^{\paren i} }_\infty\) Definition of Pointwise Addition of Real-Valued Functions
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop = 0}^k \norm {x^{\paren i} }_\infty + \sum_{i \mathop = 0}^k \norm {y^{\paren i} }_\infty\) Triangle Inequality for Real Numbers
\(\displaystyle \) \(=\) \(\displaystyle \norm x_{\map {C^k} I} + \norm y_{\map {C^k} I}\) Definition of C^k Norm

$\blacksquare$


Also see


Sources