> [!definition] > > Let $E$ be a [[Banach Space|Banach space]] and $S \subset E$, then the following are equivalent: > 1. For each $x \in S$, there exists a [[Neighbourhood|neighbourhood]] $U \in \cn^o(x)$ and $\phi \in C^k(U, \mathbb R)$ such that $S \cap U = \bracs{x \in V: \phi(x) = 0}$ and $D\phi(y) \ne 0$ for all $y \in U$. > 2. For each $x \in S$, there exists $U \in \cn^o(x)$, $V \subset E$ open, a $C^k$-isomorphism $F: U \to V$ and $\mu \in E^*$ with $\mu \ne 0$ such that $S \cap U = \bracs{x \in V: \mu \circ F = 0}$. > 3. $S$ is a submanifold of $E$ with codimension $1$. > > If the above holds, then $S$ is a **hypersurface** of class $C^k$. > > *Proof*. $(1) \Rightarrow (2)$: Let $0 \ne \mu = D\phi(x) \in E^*$ and $v \in E$ such that $\angles{v, \mu} = 1$. This induces an isomorphism > $ > T: E \to \ker(\mu) \times \mathbb R \quad x \mapsto (x - v\angles{x, \mu}, \angles{x, \mu}) > $ > Define > $ > F: U \to E \quad x \mapsto \pi_1Tx + \phi(x)v > $ > then since $\pi_1Tx \in \ker(\mu)$, > $ > \mu \circ F(x) = \phi(x)\mu(v) = \phi(x) > $ > which is $0$ if and only if $x \in S \cap U$. By the [[Inverse Function Theorem]], restricting to an appropriate neighbourhood yields a $C^k$-isomorphism. > [!definition] > > Let $E$ be a Banach and $S \subset E$ be a hypersurface of class $C^k$, then > 1. There exists a $C^{k - 1}$-[[Covector Field|covector field]] $\omega: S \to E^*$ with unit length such that $\angles{v, \omega(x)} = 0$ for every $v \in T_xS$. > 2. If $E$ is a [[Hilbert Space]], then there exists a vector field $\xi: S \to E$ with unit length such that $\angles{\eta, \xi} = \angles{\eta, \omega}$ for all $\eta: S \to E$. > > In the case of $(2)$, the vector field $\omega$ is the **normal field** to $S$. For any neighbourhood $U$ of $S$, this induces a mapping > $ > C^{k - 1}(U) \to C^{k - 1}(U) \quad f \mapsto Df(\xi) > $ > known as the **normal derivative**. In particular, on the sphere, the normal derivative field is equal to $(1/d, \cdots, 1/d)$. > > A choice of $\pm \omega$ determines an [[Orientation on Manifold|orientation]] on $S$. > > *Proof*. $(1)$: [[Gluing Lemma]]. $(2)$: [[Riesz Representation Theorem]].