§ 1. Comparison of functions on a filtered set

Let E be a set filtered by a filter with base $\mathfrak{F}$ (Gen. Top., I, p. 58); in this chapter we shall consider functions whose domain of definition is a subset of E belonging to the filter base $\mathfrak{F}$ (the subset depending on the function) and which take their values either in the field of real numbers $\mathbf{R}$, or, more generally, in a normed vector space over a valued field (Gen. Top., IX, p. 170).

In applications, E will most often be a subspace of the real space $\mathbf{R}^n$, or the extended line $\overline{\mathbf{R}}$, and $\mathfrak{F}$ will be the trace on E of the filter of neighbourhoods of a closure point of E, or else the filter of complements of relatively compact subsets of E ("neighbourhoods of the point at infinity").

It will not in general suffice to know that such a function tends to a given limit along $\mathfrak{F}$ to be able to treat all the problems of "passage to the limit along $\mathfrak{F}$" for expressions formed from this function.

For example, when the real variable x tends to $+\infty$ the three functions $x$, $x^2$ and $\sqrt{x}$ all tend to $+\infty$, but, of the expressions

the first tends to $+\infty$, the second to 1, the third to 0.

It is thus important to know not just the limit value of a function along $\mathfrak{F}$ (when this limit exists) but also the "manner" in which the function approaches its limit; in other words, one is led to classify the set of functions which tend to the same limit.

1. Comparison relations: I. Weak relations

In what follows we denote by V a normed vector space over a valued field K, and by $\mathscr{H}(\mathfrak{F}, V)$ the set of functions with values in V, each of which is defined on a subset of E belonging to the filter base $\mathfrak{F}$. The relations which we shall define between such functions have a local character relative to the filter with base $\mathfrak{F}$: let us clarify what we mean by this. If f and g are two functions from $\mathscr{H}(\mathfrak{F}, V)$, recall that the relation "there is a set $Z \in \mathfrak{F}$ such that f and g are defined and equal on Z" is an equivalence relation $R_\infty$ on $\mathscr{H}(\mathfrak{F}, V)$ (Gen. Top.*, I, p. 66). This being so, we shall say that a relation S involving a function $f$ of $\mathscr{H}(\mathfrak{F}, V)$ is of local character (along $\mathfrak{F}$) relative to $f$, if it is compatible (in $f$) with the equivalence relation $R_\infty$ (Set Theory, II, p. 117); we know that if $\tilde{f}$ is the germ of $f$ along $\mathfrak{F}$, the equivalence class of $f$ modulo $R_\infty$ (an element of the quotient set $\mathscr{H}_\infty(\mathfrak{F}, V) = \mathscr{H}(\mathfrak{F}, V)/R_\infty$), then one can derive from S, by passage to the quotient, a relation between $\tilde{f}$ and the other arguments of S, and that, conversely, every relation of this nature defines a relation of local character relative to $f$.

$^1$ In particular, in all that follows, given a function $f$ in $\mathscr{H}(\mathfrak{F}, V)$ we shall denote by $\|f\|$ the function $t \mapsto \|f(t)\|$ which belongs to $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ and is defined on the same set as $f$: we expressly draw attention to the fact that in this chapter $\|f\|$ is a function and not a number.

Given two normed vector spaces $V_1, V_2$, and two functions $f_1, f_2$ belonging to $\mathscr{H}(\mathfrak{F}, V_1)$ and $\mathscr{H}(\mathfrak{F}, V_2)$ respectively, one says that $f_1$ is dominated by $f_2$ (along $\mathfrak{F}$), and writes $f_1 \preccurlyeq f_2$ or $f_2 \succeq f_1$, if $\|f_1\| \preccurlyeq \|f_2\|$.

The relation $f_1 \preccurlyeq f_2$ is clearly of local character in $f_1$ and $f_2$; it is thus equivalent to the relation $\tilde{f}_1 \preccurlyeq \tilde{f}_2$ derived by passing to the quotient. When $f$ and $g$ are real functions one must be careful not to confuse the relations $\tilde{f} \preccurlyeq \tilde{g}$ and $\tilde{f} \leq \tilde{g}$.

Note that for every scalar $\lambda \neq 0$ the relation $f_1 \preccurlyeq f_2 \lambda$ is equivalent to $f_1 \preccurlyeq f_2$. If $f_1 \preccurlyeq f_2$ there exists a set $X \in \mathfrak{F}$ such that $f_1(x) = 0$ for every point $x \in X$ where $f_2(x) = 0$.

Examples. 1) The relation $f \preccurlyeq 1$ means that $f$ is bounded on a set of $\mathfrak{F}$.

2) For every function $f$ of $\mathscr{H}(\mathfrak{F}, V)$, and every scalar $\lambda \neq 0$, one has $f \preccurlyeq f \lambda$. 3) When $x$ tends to $+\infty$ one has $\sin^2 x \preccurlyeq \sin x$. 4) When $(x, y)$ tends to $(0, 0)$ in $\mathbf{R}^2$ one has

The following propositions are immediate consequences of def. 1:

For every scalar $\lambda \neq 0$ the relation $f \simeq g$ is equivalent to $f \simeq g \lambda$. It implies the existence of a set $X \in \mathfrak{F}$ such that that the subset of $X$ of points where $f(x) = 0$ is identical with the subset of $X$ of points where $g(x) = 0$.

2. Comparison relations: II. Strong relations

Given two normed spaces $V_1,\ V_2$ and two functions $f_1,\ f_2$ belonging to $\mathscr{H}(\mathfrak{F}, V_1)$ and $\mathscr{H}(\mathfrak{F}, V_2)$ respectively, one says that $f_1$ is negligible relative to $f_2$ (along $\mathfrak{F}$), and one writes $f_1 \ll f_2$ or $f_2 \gg f_1$, if $\|f_1\| \ll \|f_2\|$.

For every scalar $\lambda \neq 0$ the relation $f_1 \ll f_2 \lambda$ is equivalent to $f_1 \ll f_2$. The relation $f_1 \ll f_2$ implies $f_1 \preccurlyeq f_2$ but is not equivalent to it.

Note that the relation $f_1 \preccurlyeq f_2$ by no means implies the relation "$f_1 \ll f_2$ or $f_1 \asymp f_2$": one has $\sin x \preccurlyeq 1$ as $x$ tends to $+\infty$, but neither of the relations $\sin x \ll 1$ or $\sin x \asymp 1$ is true.

Examples. 1) The relation $f \ll 1$ means that $f$ tends to 0 along $\mathfrak{F}$.

2) When $\alpha$ and $\beta$ are two real numbers such that $\alpha < \beta$ one has $x^\alpha \ll x^\beta$ as $x$ tends to $+\infty$. Similarly, when $m$ and $n$ are two rational integers such that $m < n$ one has $z^m \ll z^n$ as the complex number $z$ tends to $\infty$.

3) As $x$ tends to $+\infty$ one has $x^n \ll e^x$ for every integer $n$ (III, p. 105).

4) In $\mathbf{R}^2$ one has, as $(x, y)$ tends to $(0, 0)$,

The following propositions can be deduced immediately from def. 3:

The relation $f \sim g$ implies $f \asymp g$ but is not equivalent to it.

3. Change of variable

Let $\varphi$ be a map from the set $E'$ into $E$ such that $\varphi^{-1}(\mathfrak{F})$ is a filter base on $E'$. It is clear that if $f_1, f_2$ are functions in $\mathscr{H}(\mathfrak{F}, V_1)$ and $\mathscr{H}(\mathfrak{F}, V_2)$ respectively, then $f_1 \circ \varphi, f_2 \circ \varphi$ belong to $\mathscr{H}(\varphi^{-1}(\mathfrak{F}), V_1)$ and $\mathscr{H}(\varphi^{-1}(\mathfrak{F}), V_2)$ respectively, and that the relation $f_1 \preccurlyeq f_2$ (resp. $f_1 \ll f_2$) is equivalent to $f_1 \circ \varphi \preccurlyeq f_2 \circ \varphi$ (resp. $f_1 \circ \varphi \ll f_2 \circ \varphi$).

4. Comparison relations between strictly positive functions

Let $g$ be a function in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ which is strictly positive on a set in $\mathfrak{F}$. Comparison relations featuring $g$ can be formulated in another way: the relation $f \preccurlyeq g$ is equivalent to saying that $\|f\|/g$ (which is defined on a set in $\mathfrak{F}$) is bounded on a set in $\mathfrak{F}$; the relation $f \ll g$ is equivalent to saying that $\|f\|/g$ tends to 0 along $\mathfrak{F}$. If $f$ is a function in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ the relation $f \asymp g$ means that $f/g$ is logarithmically bounded on a set in $\mathfrak{F}$, and the relation $f \sim g$ that $f/g$ tends to 1 along $\mathfrak{F}$. If $f$ is a function in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ and positive on a set in $\mathfrak{F}$, to say that $f$ and $g$ are comparable implies that $f/g$ tends to a limit (finite or equal to $+\infty$) along $\mathfrak{F}$.