> [!definition] > > Let $E, F$ be [[Banach Space|Banach spaces]], and $U \subset E$, $V \subset F$ be [[Open Set|open sets]]. A [[Function|function]] $f: U \to V$ is of **class** $C^p(U, V)$ if it is $p$-times [[Continuity|continuously]] [[Derivative|differentiable]]. > [!definition] > > By the [[Chain Rule|chain rule]], composition of $C^p$ maps are $C^p$. Therefore open subsets of Banach spaces forms a [[Category|category]], and the $p$-times continuous differentiable maps will be called $C^p$**-morphisms**. > [!theorem] > > Let $U_1, U_2, V_1, V_2$ be [[Open Set|open]] subsets of Banach spaces, and > $ > g: U_1 \times U_2 \to V_1 \times V_2 > $ > be a $C^p$ morphism. Let $a_2 \in U_2$ and $b_2 \in V_2$ such that $g(U_1 \times a_2) \subset V_1 \times b_2$, then the induced map > $ > g_1: U_1 \to V_1 \quad a_1 \mapsto \braks{g(a_1, a_2)}_1 > $ > is also a $C^p$ morphism. > > *Proof*. Since the inclusion $\iota: U_1 \to U_1 \times U_2$ and the projection $\pi_1: V_1 \times V_2 \to V_1$ are both bounded linear maps, $g_1 = \iota \circ g \circ \pi_1$ is $C^p$. > [!theorem] > > Let $E, F$ be finite-dimensional (real) vector spaces, and $U$ be an open set of a Banach space. Let > $ > f: U \times E \to F > $ > be a $C^p$-morphism such that for each $x \in U$, > $ > f_x: E \to F > $ > is linear, then the map > $ > U \to L(E, F) \quad x \mapsto f_x > $ > is a $C^p$-morphism. > > *Proof*. Since $F = \real^n$, $L(E, F) = \prod_{k}L(E, \real)$. It's sufficient to prove the claim for $F = \real$. The same works with $E$. Assuming that $E = F = \real$, then since $f_x$ is linear for each $x$, $f(x, v) = g(x)v$ for some $g: U \to \real$. As $f$ is a morphism, plugging $v = 1$ gives that $g$ is also a morphism. > [!theorem] > > Let $U$ be open in the Banach space $E$, and $F$ be a Banach space. If $f: U \to L(E, F)$ is a $C^p$-morphism, then the map > $ > U \times E \to F \quad (x, v) \to f(x) \cdot v > $ > is also a $C^p$-morphism. > > *Proof*. > $ > \begin{CD} > U \times E @>{f \times \text{Id}}>> L(E, F) \times E @>{\text{bilinear}}>> F > \end{CD} > $