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

$$X \overset{\text{def}}{=} Y .$$