1. ## bounded lineal operators

Let X,Y be two banach spaces and $T_n :X \to Y$ be linear operators that converge punctually to $T_{} :X \to Y$ Show that
$T_n \to T$ uniformly in each compact subset of X

2. Originally Posted by mms
Let X,Y be two banach spaces and $T_n :X \to Y$ be linear operators that converge punctually to $T_{} :X \to Y$ Show that
$T_n \to T$ uniformly in each compact subset of X
First, replacing $T_n$ by $T_n-T$, we may assume that $T=0$, so that $T_nx\to0$ for all $x$ in $X$.

Next, it follows from the uniform boundedness principle that $\{T_n\}$ is bounded, say $\|T_n\|\leqslant M$ for all $n$.

Now let $\varepsilon>0$. For each $x$ in $X$, there exists $N_x$ such that $\|T_nx\|<\varepsilon/2$ whenever $n\geqslant N_x$. Let $U_x$ denote the open set $\{y\in X:\|y-x\|<\frac\varepsilon{2M}\}$. Then $\|T_ny\|\leqslant \|T_nx\| + \|T_n(y-x)\|<\varepsilon$ for all $y$ in $U_x$, whenever $n\geqslant N_x$. Such sets $U_x$ form an open cover of a given compact subset of $X$, and by looking at a finite subcover you should be able to see that the pointwise convergence $T_nx\to0$ is uniform on that set.