Course 4 Book 4A: Function Spaces and Sobolev SpacesChapter 2

Baire Category and Its Consequences

Uniform boundedness, open mapping and closed graph, and why some numerical methods diverge.

19 min read · Updated Oct 2, 2026

Read with Brezis, chapter 2, "The Uniform Boundedness Principle and the Closed Graph Theorem", or Kreyszig §4.7 (category and uniform boundedness), §4.11 (numerical integration, including Pólya's theorem), §4.12–4.13 (open mapping and closed graph).

In this chapter · 6 sections

2.1When more points make things worse
2.2The Baire category theorem
2.3Uniform boundedness
2.4Open mapping, bounded inverse, closed graph
2.5History
2.6Exercises

This chapter proves three theorems that need nothing but completeness, and that say something surprisingly strong. The uniform boundedness principle: a family of bounded operators that is bounded at each point separately is bounded uniformly. The open mapping theorem: a bounded linear bijection between Banach spaces has a bounded inverse. The closed graph theorem: a linear map whose graph is closed is bounded. All three come from one lemma about complete metric spaces, the Baire category theorem, which says such a space can't be a countable union of "thin" sets.

They are used occasionally rather than constantly, but when they are used they settle a question in one line, and two of their consequences are part of everyday analysis. Weakly convergent sequences are bounded (4A.6 Weak Convergence and the Direct Method). And if an equation is uniquely solvable in Banach spaces, the solution automatically depends continuously on the data, an a-priori estimate for free. The chapter's anchor is the uniform boundedness principle at work in numerical analysis: why some natural, high-order numerical methods diverge.

By the end of this chapter you will be able to:

state and prove the Baire category theorem, and use it to show that sets are "large";
prove the uniform boundedness principle and apply it to quadrature and Fourier series;
state the open mapping, bounded inverse and closed graph theorems and use them;
explain why equally spaced high-order interpolation and quadrature can fail, and what fixes them.

When more points make things worse

In the world In use Newton–Cotes quadrature and Runge's phenomenon

The natural way to integrate a function numerically on $[0, 1]$ is to sample it at $n + 1$ equally spaced points, fit the polynomial of degree $n$ through the samples, and integrate the polynomial exactly. This gives the Newton–Cotes rules $Q_n(f) = \sum_jw_{nj}f(x_{nj})$ : the trapezoid rule for $n = 1$ , Simpson's rule for $n = 2$ . One would expect higher $n$ to be better. It isn't. For $n \geq 8$ some weights $w_{nj}$ are negative, and their total size $\sum_j|w_{nj}|$ grows explosively: computed exactly, it is $1.45$ for $n = 8$ , $3.06$ for $n = 10$ , $58$ for $n = 16$ and about $5600$ for $n = 24$ .

Large weights of both signs mean that tiny errors in the samples are hugely amplified, and George Pólya proved in 1933 that the rules then fail to converge for some continuous functions. His theorem, proved below by the uniform boundedness principle, says exactly when a sequence of quadrature rules converges for every continuous function: when it converges for polynomials and the sums $\sum_j|w_{nj}|$ stay bounded. Rules with positive weights, such as Gaussian quadrature or Clenshaw–Curtis quadrature, satisfy this automatically, which is one reason they are the methods used in practice.

The same instability appears in interpolation. Carl Runge observed in 1901 that the polynomial interpolating $f(x) = \frac{1}{1 + 25x^2}$ at equally spaced points on $[-1, 1]$ gets worse as the degree increases, swinging wildly near the ends (Figure 2.1). Computed, the worst error is about $1.9$ with $11$ points and about $60$ with $21$ . Interpolating instead at the Chebyshev points $\cos\frac{(2j+1)\pi}{2n+2}$ , which cluster near the ends, the errors are about $0.11$ and $0.015$ . Modern software for computing with functions (Trefethen's Chebfun system, for example) is built on Chebyshev interpolation for this reason.

Figure 2.1. Runge's phenomenon for

f(x) = \frac{1}{1 + 25x^2}

. Interpolation at

21

equally spaced points (orange) diverges near the ends, with errors up to about

60

(clipped here); interpolation at

21

Chebyshev points (blue) is within

0.015

f

everywhere.

The Baire category theorem

A subset of a metric space is nowhere dense if its closure has empty interior: it is not dense in any ball. A line in the plane, the integers in $\mathbb{R}$ , the Cantor set in $[0, 1]$ (3A.2 Lebesgue Measure) are nowhere dense.

Theorem 2.1 Baire category theorem

A non-empty complete metric space is not a countable union of nowhere dense sets. Equivalently: if $X = \bigcup_nF_n$ with each $F_n$ closed, then some $F_n$ contains a ball.

Proof. Let $F_n$ be closed and nowhere dense; we find a point in none of them. $F_1$ contains no ball, so there is a point outside it, and since $F_1$ is closed, a closed ball $\bar B_1$ of radius $r_1 < 1$ that misses $F_1$ . $F_2$ contains no ball, in particular not the open ball inside $\bar B_1$ , so there is a closed ball $\bar B_2 \subseteq \bar B_1$ of radius $r_2 < \tfrac12$ missing $F_2$ . Continue: nested closed balls $\bar B_1 \supseteq \bar B_2 \supseteq \cdots$ with radii $r_n < 2^{-n+1}$ , $\bar B_n$ missing $F_n$ . Their centres form a Cauchy sequence, which converges by completeness to a point lying in every $\bar B_n$ , hence in no $F_n$ .

The theorem gives a notion of size different from measure and cardinality. Countable unions of nowhere dense sets are called meagre ("of the first category"); complements of meagre sets are residual, and in a complete space they are dense. Some quick consequences:

$\mathbb{R}$ is uncountable: it is not a countable union of points, each nowhere dense.
An infinite-dimensional Banach space has no countable algebraic (Hamel) basis: it would be a countable union of finite-dimensional subspaces, each closed and nowhere dense (Exercise 2.8).
In $C([0, 1])$ with the sup norm, the functions that are differentiable at even one point form a meagre set (Banach and Mazurkiewicz, 1931). So "most" continuous functions are differentiable nowhere. Weierstrass's monster of 2B.5 Uniform Convergence and Arzelà–Ascoli is typical, not exceptional.

Meagre and null sets are both "small", but in different senses: the fat Cantor set of 3A.2 Lebesgue Measure is nowhere dense with positive measure, and $\mathbb{R}$ is the union of a null set and a meagre set (Exercise 2.9).

Uniform boundedness

Theorem 2.2 Uniform boundedness principle (Banach–Steinhaus)

Let $X$ be a Banach space, $Y$ a normed space, and $\{T_\alpha\}$ a family of bounded operators $X \to Y$ such that for each $x \in X$ , $\sup_\alpha\|T_\alpha x\| < \infty$ . Then $\sup_\alpha\|T_\alpha\| < \infty$ .

Proof. Let $F_n = \{x : \|T_\alpha x\| \leq n \text{ for all } \alpha\}$ , closed as an intersection of closed sets. By hypothesis $X = \bigcup_nF_n$ , so by Baire some $F_N$ contains a ball $B(x_0, r)$ . For $\|z\| < r$ , both $x_0$ and $x_0 + z$ lie in the ball, so $\|T_\alpha z\| \leq \|T_\alpha(x_0 + z)\| + \|T_\alpha x_0\| \leq 2N$ for every $\alpha$ . Hence $\|T_\alpha\| \leq 2N/r$ .

Pointwise bounds become uniform bounds, for free. The contrapositive is the form used to show divergence: if $\sup_\alpha\|T_\alpha\| = \infty$ , then there is a single $x$ with $\sup_\alpha\|T_\alpha x\| = \infty$ , and in fact the set of such $x$ is residual.

Theorem 2.3 Pólya's theorem on quadrature

For each $n$ let $Q_n(f) = \sum_{j=0}^{m_n}w_{nj}f(x_{nj})$ be a quadrature rule on $[a, b]$ . Then $Q_n(f) \to \int_a^bf$ for every $f \in C([a, b])$ if and only if

$Q_n(p) \to \int_a^bp$ for every polynomial $p$ , and
$\sup_n\sum_j|w_{nj}| < \infty$ .

Proof. Each $Q_n$ is a bounded linear functional on $C([a, b])$ with norm $\|Q_n\| = \sum_j|w_{nj}|$ (the norm is attained by a continuous function equal to $\operatorname{sign}w_{nj}$ at each node). Only if: if $Q_n(f)$ converges for every $f$ , then $\sup_n|Q_n(f)| < \infty$ for every $f$ , and uniform boundedness gives (2). If: given $f$ and $\varepsilon$ , take a polynomial $p$ with $\|f - p\|_\infty \leq \varepsilon$ (Weierstrass, 2B.5 Uniform Convergence and Arzelà–Ascoli). Then

\Big|Q_n(f) - \int f\Big| \leq |Q_n(f - p)| + \Big|Q_n(p) - \int p\Big| + \Big|\int(p - f)\Big| \leq \varepsilon\sup_k\|Q_k\| + o(1) + \varepsilon(b - a).

Newton–Cotes rules integrate polynomials of degree $n$ exactly, so (1) holds, but their weight sums are unbounded, so (2) fails: there is a continuous function for which they diverge. Rules with positive weights that integrate constants exactly have $\sum|w_{nj}| = \sum w_{nj} = b - a$ , so they converge for every continuous function.

Example 2.4 A continuous function whose Fourier series diverges

On the circle, the partial sum at $0$ is $S_Nf(0) = \int_0^1f(y)D_N(y)\,dy$ (2B.7 Fourier Series and the First Heat Equation), a bounded linear functional on $C(\mathbb{R}/\mathbb{Z})$ with norm $L_N = \int_0^1|D_N|$ , the Lebesgue constant. Computed, $L_N \approx 1.44, 1.88, 2.41, 2.96, 3.52$ for $N = 1, 4, 16, 64, 256$ , growing like $\frac{4}{\pi^2}\log N$ (Figure 2.2). Since $\sup_NL_N = \infty$ , uniform boundedness gives a continuous $f$ with $\sup_N|S_Nf(0)| = \infty$ : its Fourier series diverges at $0$ . Du Bois-Reymond constructed one explicitly in 1873; the Baire argument shows that such functions form a residual set. (The growth is very slow, which is why it went unnoticed for so long.)

Figure 2.2. Lebesgue constants

L_N = \int_0^1|D_N|

of the Fourier partial sums, computed (dots), against

N

on a logarithmic axis. They grow like

\frac{4}{\pi^2}\log N

(dashed): slowly, but without bound, so by the uniform boundedness principle some continuous function's Fourier series diverges.

The principle also gives the fact that 4A.6 Weak Convergence and the Direct Method needs: if $x_n \to x$ weakly (that is, $\phi(x_n) \to \phi(x)$ for every bounded linear functional $\phi$ ), then $\sup_n\|x_n\| < \infty$ . And a pointwise limit of bounded operators on a Banach space, $Tx = \lim T_nx$ , is automatically bounded.

Open mapping, bounded inverse, closed graph

Theorem 2.5 Open mapping theorem

A bounded linear map from a Banach space $X$ onto a Banach space $Y$ maps open sets to open sets.

Idea of the proof (Brezis, Theorem 2.6). Since $T$ is onto, $Y = \bigcup_nT(B_X(0, n))$ , and Baire applied to the closures gives a ball in some $\overline{T(B_X(0, n))}$ ; by linearity, $\overline{T(B_X(0, 1))}$ contains a ball around $0$ in $Y$ . A successive-approximation argument, which uses completeness of $X$ to sum a geometric series of corrections, removes the closure: $T(B_X(0, 1))$ itself contains a ball around $0$ .

Corollary 2.6 Bounded inverse theorem

A bounded linear bijection between Banach spaces has a bounded inverse.

Proof. The inverse is linear, and the preimage of an open set under $T^{-1}$ is its image under $T$ , which is open. So $T^{-1}$ is continuous.

Here is the "estimate for free". Suppose a linear differential equation $Lu = f$ has, for every $f$ in a Banach space $Y$ , exactly one solution $u$ in a Banach space $X$ , and $L : X \to Y$ is bounded. Then the solution operator is bounded: $\|u\|_X \leq C\|f\|_Y$ . Existence and uniqueness imply stability, as long as both spaces are complete. In practice one usually proves the estimate directly (because the constant matters), but the theorem guarantees the search isn't hopeless.

A second use is comparing norms. If a vector space is complete under two norms and one is bounded by a multiple of the other, then they are equivalent (Exercise 2.13). This is how different definitions of the same Sobolev or Hölder norm are shown to agree (4A.9 Sobolev Spaces, 4A.11 Hölder Spaces).

Theorem 2.7 Closed graph theorem

Let $T : X \to Y$ be linear, with $X$ and $Y$ Banach. If the graph $\{(x, Tx)\}$ is closed in $X \times Y$ (that is, $x_n \to x$ and $Tx_n \to y$ imply $y = Tx$ ), then $T$ is bounded.

Proof. The graph $G$ is a closed subspace of the Banach space $X \times Y$ (with the norm $\|x\| + \|y\|$ ), hence a Banach space. The projection $(x, Tx) \mapsto x$ is a bounded linear bijection from $G$ onto $X$ , so its inverse $x \mapsto (x, Tx)$ is bounded by Corollary 2.6, and so $\|Tx\| \leq C\|x\|$ .

To prove continuity directly, one must show that $x_n \to x$ implies that $Tx_n$ converges, and to $Tx$ . The closed graph theorem says it is enough to check the second part, assuming $Tx_n$ converges. A typical application is Exercise 2.11: a symmetric operator defined on all of a Hilbert space is automatically bounded. The contrapositive explains why the differential operators of quantum mechanics and PDE, which are unbounded, can never be defined on the whole Hilbert space, but only on a dense subspace of smooth enough functions.

Where this goes Where these theorems return

The uniform boundedness principle makes weakly convergent sequences bounded, the first step of the direct method (4A.6 Weak Convergence and the Direct Method). The open mapping theorem is behind the Fredholm alternative (4A.7 Compact Operators and Spectra): for a compact perturbation of the identity, injectivity implies bounded invertibility. In elliptic theory (6A.5 Weak Solutions and Elliptic Regularity), Fredholm theory plus the bounded inverse theorem turns uniqueness into estimates, and in the analysis of the linearised Ricci–DeTurck operator (11A.3 Short-Time Existence and Uniqueness) the existence of a bounded inverse between Hölder spaces is the key input to the contraction argument.

History

René Baire proved his category theorem for the real line in his 1899 thesis. Stefan Banach and Hugo Steinhaus proved the uniform boundedness principle in 1927, and Banach the open mapping and closed graph theorems by 1929–1932 (Juliusz Schauder gave a version of the open mapping theorem in 1930). Banach and Stefan Mazurkiewicz independently showed in 1931 that nowhere differentiable functions are generic in $C([0, 1])$ . Runge's paper on interpolation appeared in 1901, Pólya's quadrature theorem in 1933, and Hellinger and Toeplitz's theorem on symmetric operators in 1910.

Recall Where we stand

In a complete metric space, a countable union of nowhere dense sets has empty interior (Baire). Applied to Banach spaces: pointwise bounded families of operators are uniformly bounded; bounded linear bijections have bounded inverses; and linear maps with closed graphs are bounded. In numerical analysis these explain why quadrature and interpolation with unbounded weight sums diverge for some continuous functions (Pólya; Runge), and in Fourier analysis why some continuous functions have divergent Fourier series. 4A.3 Hahn–Banach and Duality turns to the dual space of bounded linear functionals, and to the Hahn–Banach theorem that provides enough of them.

Exercises

Exercise 2.8 No countable basis

Let $X$ be an infinite-dimensional Banach space. Show that a finite-dimensional subspace of $X$ is closed and nowhere dense, and deduce from Baire that $X$ has no countable Hamel basis (every vector a finite linear combination of basis vectors). Conclude that the space of polynomials, which has the countable basis $1, x, x^2, \ldots$ , is not complete under any norm.

Exercise 2.9 Small in two ways

Let $q_1, q_2, \ldots$ enumerate $\mathbb{Q}$ , let $U_k = \bigcup_n(q_n - 2^{-n-k}, q_n + 2^{-n-k})$ and $G = \bigcap_kU_k$ . Show that $G$ is null and that its complement is meagre. So $\mathbb{R}$ is the union of a null set and a meagre set: measure and category can disagree completely about which sets are small.

Solution

$m(U_k) \leq \sum_n2^{1-n-k} = 2^{1-k}$ , so $m(G) = 0$ . Each $U_k$ is open and dense (it contains $\mathbb{Q}$ ), so $\mathbb{R} \setminus U_k$ is closed and nowhere dense, and $\mathbb{R} \setminus G = \bigcup_k(\mathbb{R}\setminus U_k)$ is meagre.

Exercise 2.10 Norm of a quadrature rule

Show that $Q(f) = \sum_jw_jf(x_j)$ (distinct nodes in $[a, b]$ ) has norm exactly $\sum_j|w_j|$ as a functional on $C([a, b])$ . (Build a continuous $f$ with $|f| \leq 1$ and $f(x_j) = \operatorname{sign}w_j$ , by joining the values linearly.)

Exercise 2.11 Hellinger–Toeplitz

Let $H$ be a Hilbert space and $T : H \to H$ linear with $\langle Tx, y\rangle = \langle x, Ty\rangle$ for all $x, y$ . Show $T$ is bounded, by checking that its graph is closed. (If $x_n \to x$ and $Tx_n \to z$ , compute $\langle z, y\rangle$ for every $y$ .)

Solution

$\langle z, y\rangle = \lim\langle Tx_n, y\rangle = \lim\langle x_n, Ty\rangle = \langle x, Ty\rangle = \langle Tx, y\rangle$ for all $y$ , so $z = Tx$ .

Exercise 2.12 Closed but not everywhere defined

Let $X = C([0, 1])$ with the sup norm and $D = C^1([0, 1])$ as a subspace. Show that $T = \frac{d}{dx} : D \to X$ has closed graph (use 2B.5 Uniform Convergence and Arzelà–Ascoli's theorem on uniform limits of derivatives) but is unbounded. Which hypothesis of the closed graph theorem fails?

Solution

If $f_n \to f$ and $f_n' \to g$ uniformly, then $f \in C^1$ with $f' = g$ . Unbounded: $\sin(nx)$ . The domain $D$ is not complete in the sup norm (not closed in $X$ ).

Exercise 2.13 Rehearsal: two norms, one space

Let $X$ be complete under norms $\|\cdot\|_1$ and $\|\cdot\|_2$ , with $\|x\|_1 \leq C\|x\|_2$ for all $x$ . Show the norms are equivalent. Apply this to $C^1([0, 1])$ with $\|f\|_{C^1} = \sup|f| + \sup|f'|$ and $\|f\|_* = |f(0)| + \sup|f'|$ : show both are complete and compare them. In 4A.9 Sobolev Spaces, several equivalent norms on Sobolev spaces (with or without lower-order terms, by derivatives or by Fourier transform) are compared in exactly this way, and in 4A.11 Hölder Spaces the same is done for Hölder norms.

Solution

The identity $(X, \|\cdot\|_2) \to (X, \|\cdot\|_1)$ is a bounded bijection between Banach spaces, so its inverse is bounded: $\|x\|_2 \leq C'\|x\|_1$ . For $C^1$ : $\|f\|_* \leq \|f\|_{C^1}$ , and $\|\cdot\|_*$ is complete (a Cauchy sequence has $f_n(0)$ and $f_n'$ convergent, and $f_n = f_n(0) + \int_0^xf_n'$ converges in $C^1$ ). Directly, $\sup|f| \leq |f(0)| + \sup|f'|$ , so $\|f\|_{C^1} \leq 2\|f\|_*$ .

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2.1When more points make things worse

2.2The Baire category theorem

2.3Uniform boundedness

2.4Open mapping, bounded inverse, closed graph

2.5History

2.6Exercises

When more points make things worse

The Baire category theorem

Uniform boundedness

Open mapping, bounded inverse, closed graph

History

Exercises