I'm studying Banach Steinhaus theorem in Rudin's functional analysis (page 43-46). Theorem 2.9 says: If $X$,$Y$ are two topological vector spaces, $K$ is a compact convex set in $Y$, $\Gamma$ is a collection of continuous linear maps $X\to Y$ and the orbits $\Gamma(x)=\{\Lambda(x)|\Lambda \in \Gamma\}$ are boundend subsets of $Y$ for every $x\in K$. Then there is a bounded set $B\subset Y$ such that $\Gamma(K)\subset B$.
I understand the proof of the theorem BUT: $K$ is a locally compact Hausdorff space, so by Baire's theorem is of second cathegory. By Banach Steinhaus theorem $\Gamma|_K$ is an equicontinuous collection of linear maps, thus (theorem 2.4) $\Gamma|_K$ are equibounded. Finally since $K$ is bounded $\Gamma|_K(K)=\Gamma (K)$ is a bounded subset of $Y$.
I presume there is something wrong with this argument since I didn't use the convexity of $K$.