[Math] Classic applications of Baire category theorem

Question

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet to see.

First I found this MathOverflow problem:

Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then $f$ coincide on $[0,1]$ with some polynomial.

I found another one from Ben Green's notes:

Suppose that $f:\mathbb{R}^+\to\mathbb{R}^+$ is a continuous function with the following property: for all $x\in\mathbb{R}^+$, the sequence $f(x),f(2x),f(3x),\ldots$ tends to $0$. Prove that $\lim_{t\to\infty}f(t)=0$.

Are there any other classic problems of this type?

Answer 1

Many applications of Baire Category Theorem, such as the characterization of real polynomials you cited, follow from the strengthening of a pointwise statement
$$ \forall x \in X, \exists d \in D, \ P_d(x)$$ to an uniform statement $$ \exists d \in D, \forall x \in X, \ P_d(x), $$ where $D$ is a enumerable set and $X$ is a complete metric space.

For this, one uses the following strategy (translated from BwataBaire, a wiki in French that contains a lot of applications of Baire Category Theorem):

1. Define the set $F_d = \{x \in X \mid P_d(x) \}$.
   Since the pointwise statement is valid, we have that $ X = \bigcup_{d \in D} F_d $.
2. If we are lucky, the set $F_d$ is closed in $X$ for each $d$. By Baire Category Theorem, then there exists a $D$ such that $F_D$ has a non-empty interior.
3. By using that $\mathrm{int}(F_D)\ne\emptyset$, we try to prove that $F_D = X$, from which will follow the uniform statement.

As an example, you can try to use this strategy to prove the following:

Let $X$ be a Banach space and $f:X\rightarrow X$ a continuous linear application such that $$ \forall x \in X, \exists n \in \mathbb{N},\ f^{n}(x)=0. $$ Then $f$ is nilpotent, that is, $$ \exists n \in \mathbb{N}, \forall x \in X,\ f^n(x)=0.$$

It is interesting to remark that there also exists a lot of another common strategies to change a pointwise statement to an uniform statement. One good place to learn about them is this article from tricki.

P.S. There is also a community wiki question in math.stackexchange with a lot of this kind of applications of Baire Category Theorem: "Your favorite application of the Baire category theorem".