The Baire Category Theorem - Applied Functional Analysis

In this section we introduce the Baire category theorem. Ultimately the Baire category theorem is used to answer the question when is a subset $E \subset X$ in a topological space small, which turns out to be a fundamental tools in the study of linear bounded operators. This section and in particular the qualiatative and quantitative discussion is inspired by T. Tao Tao (2009).

Small sets in a Measure Space¶

In a measure theory setting i.e. $X = (X, \mathcal{F}, \mu)$ the “small” sets are the nullsets i.e. $E \subset X$ such that $\mu(E) = 0$ or subsets of nullsets. Countable additivity implies that countable unions of nullsets are of measure zero. This gives us results like this:

The next proposition bridges between measure theory and topology. To motivate it, recall that the Lebesgue differentiation theorem tells us that for almost every point $x \in E$ , the density of $E$ at $x$ ,

\lim_{r \to 0} \frac{\mu(E \cap B(x,r))}{\mu(B(x,r))} = 1,

(1)

equals one. Such points are called density points of $E$ — points where $E$ fills up nearly all of every sufficiently small ball. One can think of density points as measure-theoretic interior points: an interior point of $E$ is one where $E$ contains an entire ball, while a density point is one where $E$ nearly fills every small ball, even if it doesn’t contain one outright. The next proposition globalizes this local statement: if $E$ has positive measure, then $E$ must be dense in some ball in the measure-theoretic sense that it occupies an arbitrarily large fraction of that ball’s volume.

Proposition 1

Let $E$ be a measurable subset of $\mathbb{R}^d$ . Then the following are equivalent

$\mu(E) > 0$
$\forall \varepsilon > 0$ there exists a ball $B$ such that $\mu(E \cap B) \geq (1 - \varepsilon)\mu(B)$

Reading the equivalence from both sides:

(Forward): If $\mu(E) > 0$ , then $E$ is measure-theoretically dense in some ball — it nearly fills that ball.
(Reverse): If $E$ is not measure-theoretically dense in any ball — that is, in every ball, $E$ always fails to fill a definite fraction — then $\mu(E) = 0$ .

A set of measure zero is therefore nowhere dense in the measure-theoretic sense: it is never concentrated in any ball. This suggests that to formalize “smallness” in a purely topological setting, we should look for the topological analogue of this property.

Small Sets in Topological Spaces¶

To make the analogy with the measure-theoretic setting precise, we first recall what dense means in a topological space.

Remark 1

Compare this with the measure-theoretic notion from the previous section: there, $E$ is “dense in $B$ ” if $\mu(E \cap B) \approx \mu(B)$ , i.e. $E$ fills almost all the volume of $B$ . Here, $E$ is dense in $B$ if $E$ meets every open subset of $B$ , i.e. $E$ is topologically everywhere in $B$ . Volume has been replaced by topology, but the intuition is the same: $E$ is thoroughly present throughout $B$ , with no region where $E$ is absent.

This is the topological counterpart of a measure-zero set: just as a null set is never measure-theoretically dense in any ball (it never fills a large fraction of any ball), a nowhere dense set is never topologically dense in any ball (its closure contains no open set).

Note that completeness is essential: on an incomplete metric space, the Baire property can fail, and the space itself may be a countable union of nowhere dense sets. This is why completeness is such an important hypothesis throughout the theory of bounded operators. For a proof, see Hillen (2023).

Example 3 (Why do we care?)

The consequence of this theorem is that $\mathbb{R}^d$ or $\mathbb{C}^d$ cannot be covered by a countable union of proper subspaces.

Definition 4 (Proper subspaces)

$W \subset V$ is a proper subspace of $V$ such that

A subset $V$ i.e. $W \subset V$ .
Strictly smaller than $V$ .
Still a vector space.

In $\mathbb{R}^3$ examples of proper subspaces are:

The $xy$ -plane.
A line through the origin.

Baire says that we can’t write $\mathbb{R}^3$ as a countable union of these nowhere dense proper subspaces.

\mathbb{R}^2 cannot be covered by a countable union of proper subspaces (lines through the origin). — Figure 1: $\mathbb{R}^2$ cannot be covered by a countable union of proper subspaces (lines through the origin).

Remark 4

The Baire Category Theorem holds in two important classes of topological spaces:

Complete metric spaces (and hence Banach spaces, Hilbert spaces, etc.).
Locally compact Hausdorff spaces (and hence $\mathbb{R}^d$ , $\mathbb{C}^d$ , and all finite-dimensional normed spaces).

In finite dimensions these overlap: $\mathbb{R}^d$ is both complete and locally compact. In infinite dimensions, however, a Banach space is never locally compact (the closed unit ball is not compact), so completeness is the only route to the Baire property.

Consequences of the Baire Category Theorem¶

The Baire Category Theorem leads to three fundamental equivalences between the qualitative theory of continuous linear operators on Banach spaces e.g. finiteness, surjectivity to the quantitative theory e.g. estimates Tao (2009). We have already seen the first of these relationships. Recall that for a linear operator $A : X \mapsto Y$ we have that $A\ \mathrm{is\ continuous} \iff\ A\ \mathrm{is\ bounded}.$

As we saw with the examples this allows us to prove the continuity of an operator $A$ by simply establishing that it is bounded (which usually is easier).

Uniform boundedness principle that equates the qualitative operators with their quantitative boundedness.
Open mapping theorem equates qualitative solvability of a linear problem $Lu = f$ with quantitative solvability.
Closed graph theorem equates the qualitative regularity of a (weakly continuous) operator $T$ with the quantitative regularity of the operator.

Next we will explore the uniform boundedness principle, and the open mapping theorem while we leave the closed graph theorem for later.

References¶

Tao, T. (2009, February 1). 245B, Notes 9: The Baire category theorem and its Banach space consequences [Blog post]. https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/
Hillen, T. (2023). Elements of Applied Functional Analysis. https://www.math.ualberta.ca/~thillen/FA-book-June2023.pdf