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

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

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

### 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

$\blacksquare$