TMA4225 Foundations of analysis: Fall term 2014

I will on occassion depart from the notation of the book. This page is intended to list the differences.

• I use \(A\subseteq B\) for set containment (\(A\) is a subset of \(B\)) where the book uses \(A\subset B\). To me, the latter indicates strict containment, i.e., \(A\) is contained in \(B\) but the sets are not equal.
• I will use \(\mathbb{R}\), \(\mathbb{Q}\), \(\mathbb{Z}\), and \(\mathbb{N}\) for the sets of real numbers, rational numbers, integers, and natural numbers (which start with 1, not 0) respectively. The book uses calligraphic letters \(\mathcal{R}\), \(\mathcal{Q}\), \(\mathcal{Z}\), and \(\mathcal{N}\) for these sets. Oddly, they do use \(\mathbb{C}\) for the complex numbers, as will I.
• If \(\mathcal{C}\) is a set of sets, I just might write \(\bigcup\mathcal{C}\) and \(\bigcap\mathcal{C}\) for the union and intersection of all the members of this set. The book uses the more complicated, but possibly clearer, notation \(\bigcup_{A\in\mathcal{C}}A\) and \(\bigcap_{A\in\mathcal{C}}A\) for these sets. To be clear, \(x\in\bigcup\mathcal{C}\) if and only if \(x\) belongs to some member of \(\mathcal{C}\), while \(x\in\bigcap\mathcal{C}\) if and only if \(x\) belongs to every member of \(\mathcal{C}\).
• I tend to dislike the prefix “non-”, so I avoid it as much as I can. Thus, I prefer “increasing” over “non-decreasing”. I will then say “strictly increasing” if that is what I mean; in particular, a constant function is increasing in my terminology, but not strictly so. Similarly for “decreasing” versus “non-increasing”. I find the negation in the “non-” constructions confusing.
• I use the Iverson bracket, which I also like to call the indicator bracket: If \(S\) is some statement, then \[ [S]=\begin{cases} 1&\text{if $S$ is true,}\\0&\text{if $S$ is false.}\end{cases}\] I even use it to denote the characteristic function (also called the indicator function of a set \(E\): \[ [E](x)=[x\in E]=\begin{cases} 1&\text{if $x\in E$,}\\0&\text{if $x\notin E$.}\end{cases}\] This function is more commonly written \(\chi_E\).
• I am going to depart a little from the book on the notation for sections of functions (in §6.3). So if \(f\) is a function on \(\colon\Gamma\times\Lambda\), then for each \(x\in\Gamma\), I use either of the two notations \(f(x,{\cdot})\) or \(y\mapsto f(x,y)\) for the \(\Gamma\)-section at \(x\in\Gamma\), and of course the similar \(f({\cdot},y)\) or \(x\mapsto f(x,y)\) for the \(\Lambda\)-section at \(y\in\Lambda\).
  Both of these notations are quite standard in mathematics. To check that you understand thems, try to make sense of the identities \(f({\cdot},y)(a)=f(a,y)=\bigl(x\mapsto f(x,y)\bigr)(a)\). (I will probably never write such a thing, though.)
  I will shy away from sections of sets, but will just use the corresponding sections of characteristic functions instead: For instance, if \(E\subseteq\Gamma\times\Lambda\) then \([E]({\cdot},y)\) is clearly the characteristic function of some subset of \(\Gamma\), since it takes only the values \(0\) and \(1\). It might be better written using dot notation as \([({\cdot},y)\in E]\).
• The book's definition of absolute continuity for functions is backwards: Sensible people take that definition as a theorem. The proper definition of absolute continuity is the “equivalent condition” given in Prop. 8.6 on p. 294.

That completes the list, unless I think of something I forgot to mention here.