C^k Norm is Norm

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 $\struct {\map {\CC^k} I, +, \, \cdot \,}_\R$.

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

Then:

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:

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

Also see

 * Definition:C^k Norm