Definition:Space of Continuous Functions of Differentiability Class k

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Let $X, Y$ be normed vector spaces.

Let $f : X \to Y$ be a mapping of a differentiability class $k \in \N_{>0}$ in the sense of Frechet.

Then the set of all such mappings $f$ is known as continuous function space of differentiability class k and is denoted by $C^k \paren {X,Y}$:

$C^k \paren {X, Y} = \set {f : X \to Y}$