Banach-Steinhaus Theorem/Topological Vector Space

Theorem
Let $\GF \in \set {\R, \C}$.

Let $X$ and $Y$ be topological vector spaces over $\GF$.

Let $\Gamma$ be a set of continuous linear transformations $X \to Y$.

Let $B$ be the set of all $x \in X$ such that:
 * $\map \Gamma x = \set {T x : T \in \Gamma}$

is von Neumann-bounded in $Y$.

Suppose that $B$ is not meager in $X$.

Then $B = X$ and $\Gamma$ is equicontinuous.

Proof
Let $W$ be an open neighborhood of ${\mathbf 0}_Y$.

From Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods, there exists an open neighborhood $V$ of ${\mathbf 0}_Y$ such that:
 * $V + V \subseteq W$

From Disjoint Compact Set and Closed Set in Topological Vector Space separated by Open Neighborhood: Corollary, there exists a further open neighborhood $U$ of ${\mathbf 0}_X$ such that:
 * $U^- \subseteq V$

so that:
 * $U^- + U^- \subseteq W$

where $U^-$ denotes the closure of $U$ in $Y$.

From Open Neighborhood of Origin in Topological Vector Space contains Balanced Open Neighborhood and Set Closure Preserves Set Inclusion we can take $U$ to be balanced.

Let:
 * $\ds E = \bigcap_{T \in \Gamma} T^{-1} \sqbrk {U^-}$

Now let $x \in B$.

Since $\map \Gamma x$ is von Neumann-bounded in $Y$, there exists $n \in \N$ such that:
 * $\map \Gamma x \subseteq n U$

Then we have:
 * $T x \in n U$ for each $T \in \Gamma$.

That is:
 * $x \in T^{-1} \sqbrk {n U}$

so that:
 * $x \in T^{-1} \sqbrk {n U^-}$

for each $T \in \Gamma$.

Then we have:

So we may conclude that:
 * $\ds B \subseteq \bigcup_{n \mathop = 1}^\infty n E$

We argue that $n E$ is not meager in $X$ for some $n \in \N$.

that $n E$ is meager in $X$ for each $n \in \N$.

Then $\paren {n E} \cap B$ is meager in $X$ for each $n \in \N$ by Subset of Meager Set is Meager Set, while:
 * $\ds B = \bigcup_{n \mathop = 1}^\infty \paren {\paren {n E} \cap B}$

From Countable Union of Meager Sets is Meager, we have that $B$ is meager, contrary to our assumption that it is not.

So we have that $n E$ is meager in $X$ for some $n \in \N$.

From