Function (notation)

This page describes the notation of how we use functions for information on what a function is, see function

Basics

• [ilmath]f:X\rightarrow Y[/ilmath] - the most basic form, here [ilmath]f[/ilmath] is a relation that associates with each [ilmath]x\in X[/ilmath] a [ilmath]y\in y[/ilmath]. We write this as [ilmath]y=f(x)[/ilmath]

Abuses of notation

• Tuples: sometimes we will write [ilmath]f:(X,A)\rightarrow(Y,B)[/ilmath], this simply means that [ilmath]f:X\rightarrow Y[/ilmath] where [ilmath]X[/ilmath] is some sort of space (with structure [ilmath]A[/ilmath]) and [ilmath]Y[/ilmath] is some sort of space with a structure [ilmath]B[/ilmath].
  • Possible misinterpretation:

    [ilmath]f:X\times A\rightarrow Y\times B[/ilmath] denotes a function, [ilmath]f[/ilmath] that takes ordered pairs, [ilmath](x,a)\in X\times A[/ilmath] to ordered pairs, [ilmath](y,b)\in Y\times B[/ilmath], this notation clearly operates on sets (as it uses the Cartesian product) keeping with the convention of the thing either side of the [ilmath]\rightarrow[/ilmath] is a set. So the notation [ilmath]f:(X,A)\rightarrow(Y,B)[/ilmath] for [ilmath]f[/ilmath]'s input being a tuple, [ilmath](x,a)[/ilmath] is absurd because:

    1. It is another notation for something we already have
    2. It violates the "sets being either side of the arrow" thing ([ilmath]A\times B[/ilmath] is a set, [ilmath](\cdot,\cdot)[/ilmath], even if considered as an Ordered pair does not "evaluate" to something useful when it comes to relations.

  • Warnings:
    1. Sometimes [ilmath]f:(X,A)\rightarrow(Y,B)[/ilmath] denotes that [ilmath]f:X\rightarrow Y[/ilmath] with the additional information of [ilmath]f\vert_A:A\rightarrow B[/ilmath], or more simply is to say that [ilmath]f:X\rightarrow Y[/ilmath] with the additional statement: [ilmath]f(A)\subseteq B[/ilmath]
       • Use this if [ilmath](X,A)[/ilmath] has not previously been declared as some sort of space.
       • I've only ever seen this used in one book - Fundamentals of Algebraic Topology by Steven H. Weintraub
  • Examples:
    1. Let [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath] be topological spaces, let [ilmath]f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})[/ilmath] be a continuous map...
       • Here the tuples (as usual) help the reader/writer keep track of spaces, in this case the topologies on [ilmath]X[/ilmath] and [ilmath]Y[/ilmath]
       • This example extends to measurable spaces, vector spaces and many more.