A sequence of functions can behave well at every individual input while behaving badly as a family. For instance, the functions $f_k:[0,1] \to \mathbb{R}$ given by $f_k(x)=x^k$ never exceed $1$ at any fixed $x$, and for each fixed $x$ the numerical sequence $(f_k(x))_{k=1}^{\infty}$ has a finite bound. Yet the family changes shape dramatically near $x=1$, and its pointwise limit is discontinuous. Pointwise boundedness is the first vocabulary for separating these two ideas: boundedness at each input, and boundedness controlled by one constant for all inputs.

This distinction matters because many compactness and convergence arguments begin with a sequence $(f_k)_{k=1}^{\infty}$ and ask what can be salvaged from partial information. Pointwise boundedness is local in the variable: each $x$ is allowed to have its own bound. It is weaker than uniform boundedness, weaker than boundedness in most function-space norms, and often too weak by itself to control continuity, integrability, or convergence. Its strength is that it is exactly the hypothesis naturally produced by pointwise estimates and exactly the conclusion naturally tested by evaluation at a point.

The guiding question for this chapter is: when does controlling the numerical sequence $(f_k(x))_{k=1}^{\infty}$ separately for each $x$ give meaningful control over the original sequence of functions? The answer depends on what extra structure is present. On a bare set, pointwise boundedness is only a family of independent scalar bounds. On a compact [metric space](/page/Metric%20Space) with equicontinuity, it becomes part of Arzela-Ascoli. In a [Banach space](/page/Banach%20Space), it is the hypothesis of the [uniform boundedness principle](/theorems/549) when the functions are bounded linear operators.

[example: A Bounded Sequence at Each Point with a Discontinuous Limit] Let $f_k:[0,1]\to\mathbb{R}$ be defined by $f_k(x)=x^k$. For every $x\in[0,1]$ and every $k\in\mathbb{N}$, we have $0\le x\le 1$, so $0\le x^k\le 1^k=1$. Hence \begin{align*} |f_k(x)|=|x^k|=x^k\le 1. \end{align*} Thus the sequence $(f_k)_{k=1}^{\infty}$ is pointwise bounded on $[0,1]$, with the valid choice $M_x=1$ for every $x\in[0,1]$. Now fix $x\in[0,1)$. Since $0\le x<1$, the geometric powers satisfy $\lim_{k\to\infty}x^k=0$, so \begin{align*} \lim_{k\to\infty}f_k(x)=\lim_{k\to\infty}x^k=0. \end{align*} At the endpoint, \begin{align*} f_k(1)=1^k=1 \end{align*} for every $k$, so \begin{align*} \lim_{k\to\infty}f_k(1)=1. \end{align*} Therefore the pointwise limit $f:[0,1]\to\mathbb{R}$ is given by $f(x)=0$ for $0\le x<1$ and $f(1)=1$. This limit is not continuous at $1$: for every $\delta>0$, choose $x\in[0,1)$ with $1-\delta<x<1$, for instance $x=1-\delta/2$ if $\delta\le 2$ and $x=1/2$ if $\delta>2$. Then $|x-1|<\delta$, but \begin{align*} |f(x)-f(1)|=|0-1|=1. \end{align*} Thus pointwise boundedness, even together with pointwise convergence, does not preserve continuity. [/example]

The example also shows why the word "pointwise" must be taken seriously. The bound near $x=1$ is not supplied by a stable shape of the graphs; it is supplied separately at each point. The rest of the chapter develops this separation between pointwise information, uniform information, and compactness information.

## Definition

The central property is not a new kind of function, but a way of bounding a whole sequence after an input has been fixed. The definition deliberately allows the bound to depend on the input, because this is exactly what separates pointwise information from uniform information.

[definition: Pointwise Bounded Sequence] Let $X$ be a set and let $(f_k)_{k=1}^{\infty}$ be a sequence of functions $f_k:X \to \mathbb{R}$ or $f_k:X \to \mathbb{C}$. The sequence $(f_k)_{k=1}^{\infty}$ is pointwise bounded on $X$ if for every $x \in X$ there exists a constant $M_x < \infty$ such that \begin{align*} |f_k(x)| &\le M_x \end{align*} for every $k \in \mathbb{N}$. [/definition]

The parent object behind this definition is an ordinary sequence. A sequence records a countable ordered list, and in analysis the entries may be numbers, functions, operators, sets, or other mathematical objects.

[definition: Sequence] Let $A$ be a set. A sequence in $A$ is a function $a:\mathbb{N}\to A$, usually written as $(a_k)_{k=1}^{\infty}$. [/definition]

To speak about pointwise boundedness, the entries cannot merely be unrelated objects: they must be functions with a shared domain, so that the same input $x$ can be evaluated across the whole list. The next definition isolates that common-domain structure before we ask whether the resulting scalar sequence is bounded at each point.

[definition: Sequence of Functions] Let $X$ be a set and let $Y$ be a set. A sequence of functions from $X$ to $Y$ is a sequence $(f_k)_{k=1}^{\infty}$ such that each entry is a function $f_k:X \to Y$. [/definition]

The subscript in $M_x$ is the whole point of the definition. A different point may require a different bound. In quantifier form, the definition says

\begin{align*} \forall x \in X\,\exists M_x < \infty\,\forall k \in \mathbb{N}: |f_k(x)| \le M_x. \end{align*}

Changing the order of the first two quantifiers gives a stronger condition, which is uniform boundedness.

A reader should also notice what the definition does not require. It does not require any topology on $X$, any continuity of the functions, any convergence of the sequence, or any measurability. It is a property of evaluations and scalar bounds.

## Quantifiers and First Examples

### The Quantifier Swap

The definition of pointwise boundedness has two separate choices built into it: first choose the point $x$, and only then choose a bound for the resulting numerical sequence. The most common mistake is to reverse that order and act as though one bound had already been chosen before $x$ was known. Before looking at examples, it is useful to name the stronger condition obtained by making that reversal legitimate.

### Uniform Bounds