C k map between Banach spaces

Let $X$ and $Y$ be real Banach spaces and let $U\subseteq X$ be open. All derivatives below are Fréchet derivatives (Fréchet derivative between Banach spaces), and all continuity is with respect to the norm metrics (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form).

• $f:U\to Y$ is of class $C^0$ when $f$ is continuous on $U$.
• $f$ is of class $C^1$ when $f$ is differentiable on $U$ and its derivative map $Df:U\to \mathcal{B}(X,Y)$ is continuous for the operator norm on $\mathcal{B}(X,Y)$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators).
• For $k\ge 1$ define the spaces of iterated derivative values by ${\mathcal{B}}_0:=Y,{\mathcal{B}}_k:=\mathcal{B}(X,{\mathcal{B}}_{k-1})\mathrm{.}$ Each ${\mathcal{B}}_k$ is a real Banach space for the operator norm, by induction on $k$ from If (Y) is Banach then (\mathcal B(X,Y)) is Banach. Then $f$ is of class $C^k$ when $f$ is of class $C^{k-1}$, the map $D^{k-1}f:U\to {\mathcal{B}}_{k-1}$ is differentiable on $U$, where $D^{0}f:=f$, and the $k$-th derivative $D^{k}f:U\to {\mathcal{B}}_k$ is continuous for the operator norm on ${\mathcal{B}}_k$. We write $D^{k}f(x)$ for the value at $x$.
• $f$ is of class $C^{\mathrm{\infty }}$ when $f$ is of class $C^k$ for every $k\ge 0$.

For open sets $U\subseteq X$ and $V\subseteq Y$, a map $f:U\to V$ is a $C^k$ diffeomorphism when it is a bijection, $f$ is of class $C^k$ and $f^{-1}:V\to U$ is of class $C^k$; likewise with $C^{\mathrm{\infty }}$ in place of $C^k$.

• Currying and the joint norm. The space ${\mathcal{B}}_k$ of the definition is canonically identified with the space of bounded $k$-linear maps $X^k\to Y$ by currying, $T\leftrightarrow (h_1,\dots ,h_k)\mapsto (\cdots ((T{h}_1)h_2)\cdots )h_k$, and under this identification the operator norm on ${\mathcal{B}}_k$ is the least constant $C$ with $\mathrm{\parallel }T(h_1,\dots ,h_k)\mathrm{\parallel }\le C\mathrm{\parallel }{h}_1\mathrm{\parallel }\cdots \mathrm{\parallel }{h}_k\mathrm{\parallel }$. Only the iterated-operator description is used on this page, so the identification is recorded as a reading convention rather than developed.

• Continuity of the $k$-th derivative is an operator-norm condition. It is the continuity of $x\mapsto D^{k}f(x)$ in the norm of ${\mathcal{B}}_k$; this is strictly stronger than the pointwise continuity of each scalar or vector map $x\mapsto D^{k}f(x)(h_1,\dots ,h_k)$ for fixed $h_i$. The definition uses the operator norm, exactly as the sources do, and the inverse function theorem below is proved for this notion.

• $C^1$ has the chain rule, so the class is stable under composition for $k=1$. If $f:U\to V$ and $g:V\to W$ are of class $C^1$ with $V$ open, then $g\circ f$ is of class $C^1$: it is differentiable by the chain rule (Chain sum product and composition rules for Banach derivatives), its derivative is $D(g\circ f)(x)=Dg(f(x))Df(x)$, and this is continuous in $x$ because $x\mapsto (Dg(f(x)),Df(x))$ is continuous and operator multiplication is a jointly continuous bilinear operation (For a bilinear map, boundedness is equivalent to joint continuity, Composition satisfies |ST|\le|S|,|T|). The corresponding statement for $C^k$ with $k\ge 2$ is an induction of the same shape, using the higher chain rule; it is not developed here because no item on this page beyond the $C^1$ statements consumes it, and nothing below asserts it.

• Linear and affine maps. A bounded linear $T:X\to Y$ is of class $C^{\mathrm{\infty }}$ on $X$, all of whose derivatives equal $T$ at the first step and $0$ afterwards; constant maps are of class $C^{\mathrm{\infty }}$ with derivative $0$. Consequently a $C^k$ map followed or preceded by a bounded linear isomorphism between open sets is again of class $C^k$, a reduction used in the inverse function theorem.

• Where this definition is consumed. The inverse and implicit function theorems state their conclusions in this class of regularity, and the countable-base Banach manifolds defined later on this page use exactly this notion for their transition maps and coordinate representatives.