# Notation

\(\newcommand{\R}{\mathbb{R}}\) \(\newcommand{\Z}{\mathbb{Z}}\) \(\newcommand{\RR}{\mathbb{R}}\) \(\newcommand{\C}{\mathbb{C}}\) \(\newcommand{\N}{\mathbb{N}}\)

## Notation¶

The notation mostly follows what is standard in mathematics and physics; anything that might be unfamiliar to people with that background is described below.

### Types¶

should be read as "3, which is an integer". Similarly

should be read as "\(f\), which is a function from \(A\) to \(B\)".

Whenever you see \(a : b\), it is a way of indicating what *type* of thing \(a\) is, or equivalently, what space it "lives" in.

## Note

\(3 \in \mathbb{Z}\) is used in roughly the same way here ("3 is in the set of integers"). For technical reasons that are not important here, this is not quite the same as \(3 : \mathbb{Z}\).

### Partial application¶

Suppose I have a function \(f\) of type \(\R \to (\R \to \R)\), so a function which takes a real number and returns a function from \(\R\) to \(\R\). Then it makes sense to write e.g. \(f(4)(5)\), where \(4\) is the input to \(f\), and \(5\) is the input to \(f(4)\), which is itself a function.

### Equality¶

should be read as "x is defined to mean \(3\)".

### Lambda calculus¶

\(\lambda x: f(x)\) denotes the function \(x \mapsto f(x)\).

## Einstein summation¶

Einstein notation is used by default.