Requirements for the principle of uniform boundedness The version of the principle of uniform boundedness as we stated it in the lecture seems wrong to me in multiple points. Here is how I would state and proof the principle in the terms we used in the lecture
Theorem:
Let $\Gamma\subseteq L(E,F)$ be a subset of the linear, continuous functions from the normed space $E$ to the normed space $F$, $A\subseteq E$ a set of second category and $\sup_{T\in\Gamma}\|Tx\|<\infty\forall x\in A$
$\Rightarrow \|T\|\leq M$ for some $M>0$ and all $T\in\Gamma$ 
Proof:
$V_n=\{x\in E| \sup_{T\in\Gamma}\|Tx\|\leq n\}$
$\Rightarrow A\subseteq \bigcup_{n\in \mathbb{N}}V_n$
$V_n$ is closed, $A$ of 2nd category $\Rightarrow \exists m\in\mathbb{N}:V_m^\circ\neq \emptyset$
$\Rightarrow\exists \epsilon>0, x_0\in V_m^\circ: U_\epsilon(x_0)\subseteq V_m^\circ$
$\Rightarrow U_\epsilon(0)\subseteq \frac{1}{2}U_\epsilon(x_0)+\frac{1}{2}U_\epsilon(-x_0)\subseteq V_m$ since $V_m$ is absolutely convex
$\Rightarrow \|T\|=\sup_{\|x\|\leq 1}\|Tx\|=\frac{2}{\epsilon}\sup_{\|x\|\leq\frac{\epsilon}{2}}\|Tx\|\leq \frac{2m}{\epsilon}$ for all $T\in\Gamma$ q.e.d. 
The theorem as it is stated in the lecture makes the requirements that $E$ is a Banach space and $F$ a complete space, in the proof, the category theorem of Baire is used to ensure that a $V_m^\circ\neq \emptyset$ exists. I get that if we require for $E$ to be a Banach space and $A=E$, we need Baire's theorem to make sure that such a $m$ exists. However if $A$ is of 2nd category this follows because otherwise $A$ would be a meagre set.. am I missing something?
 A: Your version of the theorem, and your proof, are both correct. Indeed the completeness (or not) of $F$ is entirely irrelevant for the uniform boundedness of the family $\Gamma$, and it suffices that you have pointwise boundedness on a non-meagre set.
But it is not necessarily wrong to state a theorem in lesser generality than possible. If the more general version is never needed in the lecture, is there a compelling reason to state the more general version?
In practice, with the possible exception of experts doing nitty-gritty work, one uses the normed-space variant of the uniform boundedness principle only for Banach spaces (I'm not even sure whether a normed space is of the second category in itself if and only if it is a Banach space, or whether there are incomplete normed spaces that are of the second category in themselves), and where pointwise boundedness on the whole space is given or verifiable with no more effort than pointwise boundedness on a non-meagre set.
Stating the more general version would come at the cost of confusing the less topologically versed students more. It can be a sensible choice to opt for the less general version.
