C^k Function Space is Banach Space

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 $\struct {\map {C^k} I, +, \, \cdot \,}_\R$ equipped with $\norm {\, \cdot \,}_{\map {C^k} I}$ is a Banach space.

Proof

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Also see

• Definition:C^k Norm
• C^k Norm is Norm

Sources

• Weisstein, Eric W. "$C^k$ Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/C-kFunction.html