7.1: A- Basic Notation and Terminology

We quickly review some basic notation used in this book that is perhaps not described elsewhere. We use \(\mathbb{C}\), \(\mathbb{R}\) for complex and real numbers, and \(i\) for imaginary unit (a square root of \(-1\)). We use \(\mathbb{N} = \{ 1,2,3, \ldots \}\) for the natural numbers, \(\mathbb{N}_0 = \{ 0,1,2,3, \ldots \}\) for the zero-based natural numbers, and \(\mathbb{Z}\) for all integers. When we write \(\mathbb{C}^n\) or \(\mathbb{R}^n\) we implicitly mean that \(n \geq 1\), unless otherwise stated.

We denote set subtraction by \(A \setminus B\), meaning all elements of \(A\) that are not in \(B\). We denote complement of a set by \(X^c\). The ambient set should be clear. So, for example, if \(X \subset \mathbb{C}\) naturally, then \(X^c = \mathbb{C} \setminus X\). Topological closure of a set \(S\) is denoted by \(\overline{S}\), its boundary is denoted by \(\partial S\). If \(S\) is a relatively compact subset of \(X\) (its closure in \(X\) is compact) or compact, we write \(S \subset \subset X\).

A function with domain \(X\) and codomain \(Y\) we denote by \(f \colon X \to Y\). The direct image of \(S\) by if is \(f(S)\). The notation \(f^{-1}\) for the inverse image of sets and single points. When \(f\) is bijective (one-to-one and onto), we use \(f^{-1}\) for the inverse mapping. So \(f^{-1}(T)\) for a set \(T \subset Y\) denotes the points of \(X\) that \(f\) maps to \(T\). For a point \(q\), \(f^{-1}(q)\) denotes the points that map to \(q\), but if the mapping is bijective, then it means the unique point mapping to \(q\). To define a function without giving it a name, we use \[x \mapsto F(x),\] where \(F(x)\) would generally be some formula giving the output. The notation \(f|_S\) is the restriction of \(f\) to \(S\): a function \(f|_S \colon S \to Y\) such that \(f|_S(x) = f(x)\) for all \(x \in S\). A function \(f \colon U \to \mathbb{C}\) is compactly supported if the support, that is the set \(\overline{\{ p \in U : f(p) \not= 0 \}}\), is compact. If \(f(x) = g(x)\) for all \(x\) in the domain, we write \[f \equiv g ,\] and we say that \(f\) and \(g\) are identically equal. The notation \[f \circ g\] denotes the composition defined by \(x \mapsto f\bigl(g(x)\bigr)\).

To define \(X\) to be \(Y\) rather than just show equality, we write \[X \overset{\text{def}}{=} Y .\]