2 Half Spaces

Definition 2.1 (Half spaces) The half space of dimension \(n\) is given as \[ ℍ^n := \{x ∈ ℝ^n: π^n(x) ≥ 0\} \] where \(π^n(x^1, …, x^n) = x^n\) is the \(n\)-th coordinate map.

Definition 2.2 (Smoothness in half spaces) Let \(S\) be an open subset of \(\mathbb H^n\), and \(f: S \to \mathbb R^m\). The function \(f\) is said to be smooth at \(p \in S\) in \(\mathbb H^n\) if it admits an smooth extension in a neighborhood of \(p\) relative to \(\mathbb R^n\). That is,

1. There exists \(g: N \to \mathbb R^m\) defined on an open set \(N \in \mathbb R^m\) that contains \(p\)
2. \(f(x) = g(x)\) for all \(x \in N \cap S\).

If so, the differential of \(f\) at \(p\) is defined as the same of that of \(g\).

In addition, function \(f\) is said to be smooth in \(\mathbb H^n\) (or simply smooth) if \(f\) is smooth at every point in \(S\).

Theorem 2.1 A function \(f: S \to \mathbb R^m\) is smooth in \(\mathbb H^n\) if and only if it admits an extension in a neighborhood of \(S\) that is smooth in \(\mathbb R^n\).

Proposition 2.1

1. The composition of smooth functions are smooth.
2. Chain rule.