Let X,Y be two banach spaces and $\displaystyle T_n :X \to Y$ be linear operators that converge punctually to $\displaystyle T_{} :X \to Y$ Show that
$\displaystyle T_n \to T$ uniformly in each compact subset of X
First, replacing $\displaystyle T_n$ by $\displaystyle T_n-T$, we may assume that $\displaystyle T=0$, so that $\displaystyle T_nx\to0$ for all $\displaystyle x$ in $\displaystyle X$.
Next, it follows from the uniform boundedness principle that $\displaystyle \{T_n\}$ is bounded, say $\displaystyle \|T_n\|\leqslant M$ for all $\displaystyle n$.
Now let $\displaystyle \varepsilon>0$. For each $\displaystyle x$ in $\displaystyle X$, there exists $\displaystyle N_x$ such that $\displaystyle \|T_nx\|<\varepsilon/2$ whenever $\displaystyle n\geqslant N_x$. Let $\displaystyle U_x$ denote the open set $\displaystyle \{y\in X:\|y-x\|<\frac\varepsilon{2M}\}$. Then $\displaystyle \|T_ny\|\leqslant \|T_nx\| + \|T_n(y-x)\|<\varepsilon$ for all $\displaystyle y$ in $\displaystyle U_x$, whenever $\displaystyle n\geqslant N_x$. Such sets $\displaystyle U_x$ form an open cover of a given compact subset of $\displaystyle X$, and by looking at a finite subcover you should be able to see that the pointwise convergence $\displaystyle T_nx\to0$ is uniform on that set.