C^k Norm is Norm

Theorem

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

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

Let $x \in \map {C^k} I$ 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 $\struct {\map {C^k} I, +, \, \cdot \,}_\R$.

Proof

Positive definiteness

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

Then:

\(\ds \norm x_{\map {C^k} I}\)

\(=\)

\(\ds \sum_{i \mathop = 0}^k \norm {x^{\paren i} }_\infty\)

\(\ds \)

\(\ge\)

\(\ds \sum_{i \mathop = 0}^k 0\)

Supremum Norm is Norm, Norm Axiom $\text N 1$: Positive Definiteness

\(\ds \)

\(=\)

\(\ds 0\)

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

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

Hence:

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

That is:

$\norm x_\infty = 0$

By Supremum Norm is Norm and Norm Axiom $\text N 1$: Positive Definiteness:

$\forall t \in I : \map x t = 0$

Positive homogeneity

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

Then:

\(\ds \norm {\alpha x}_{\map {C^k} I}\)

\(=\)

\(\ds \sum_{i \mathop = 0}^k \norm {\paren {\alpha x}^{\paren i} }_\infty\)

Definition of C^k Norm

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop = 0}^k \norm {\alpha x^{\paren i} }_\infty\)

Definition of Pointwise Scalar Multiplication of Real-Valued Functions

\(\ds \)

\(=\)

\(\ds \size \alpha \sum_{i \mathop = 0}^k \norm {x^{\paren i} }_\infty\)

Supremum norm on continuous real-valued functions is a norm: positive homogeneity

\(\ds \)

\(=\)

\(\ds \size \alpha \norm x_{\map {C^k} I}\)

Definition of C^k Norm

Triangle inequality

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

\(\ds \norm {x + y}_{\map {C^k} I}\)

\(=\)

\(\ds \sum_{i \mathop = 0}^k \norm {\paren {x + y}^{\paren i} }_\infty\)

Definition of Pointwise Addition of Real-Valued Functions

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop = 0}^k \norm {x^{\paren i} + y^{\paren i} }_\infty\)

Definition of Pointwise Addition of Real-Valued Functions

\(\ds \)

\(\le\)

\(\ds \sum_{i \mathop = 0}^k \norm {x^{\paren i} }_\infty + \sum_{i \mathop = 0}^k \norm {y^{\paren i} }_\infty\)

Supremum Norm is Norm, Norm Axiom $\text N 3$: Triangle Inequality

\(\ds \)

\(=\)

\(\ds \norm x_{\map {C^k} I} + \norm y_{\map {C^k} I}\)

Definition of C^k Norm

$\blacksquare$

Also see

• Definition:C^k Norm

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed Spaces