In Rudin's Functional Analysis text, there's a theorem on page 45:
If $\Gamma$ is a collection of continuous linear maps from a F-space space $X$ to a topological vector space $Y$, and if $\{Tx: T \in \Gamma \}$ is bounded in $Y$ for each $x$ in $X$, then $\Gamma$ is equicontinuous.
I don't understand the following conclusion that Rudin makes from this:
If $X$ and $Y$ are both Banach, and if $\sup_{T \in \Gamma}\|Tx\| < \infty$ for all $x$ in $X$, then there exists $M<\infty$ such that $\|Tx\| \le M$ if $\|x\|≤1$ and $T$ in $G$.
How does this follow?
Furthermore, he claims that this implies $\|Tx\| \le M\|x\|$ if $x \in X$ and $T \in \Gamma$. How?
Let $N_x=\sup_{T \in \Gamma}||Tx|| < \infty$, this implies that $\{\|T(x)\|<N_x,T\in\Gamma\}$ so we deduce that $\{T(x),T\in \Gamma\}$ is bounded for every $x$ and $\Gamma$ is equicontinuous.
This implies that there exists $c$ such that $\|x\|< 2c$ implies that $\|T(x)\|<1$ for every $T\in \Gamma$, if $\|x\|=1, \|cx\|=c<2c$, we deduce that $\|T(x)\|={1\over c}\|T(cx)\|\leq {1\over c}$.