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".

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".

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".

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".