Let $E$ be a Banach space, and let $A(E)$ denote the closure of the finite rank opertors on $E$. Let $(S_\alpha)$ be a bounded net of operators on $E$ such that $S_\alpha T\to T$ for all $T\in A(E)$. Then
1- $S_\alpha\to I$ in the strong operator topology.
2- Use part 1 to prove that $S_\alpha\to I $ uniformly on all compact sets of $E$ .
I'm studying the relation between the existence of bounded approximate identity for $A(E)$ and some properties of the space $E$. In one of the proofs the above fact has been used, I'm trying to see why this is true but I'm stuck. Any ideas please! Thank you in advance.
Given any $x\in E$, there exists a rank-operator $T$ with $Tx=x$ (use Hahn-Banach to construct a bounded functional $f$ with $f(x)=1$, and then define $Ty=f(y)x$). Then $$ S_\alpha x=S_\alpha Tx\to Tx=x. $$ This shows part 1.
For part 2, let $X\subset E$ be compact. Fix $\varepsilon>0$. Then there exist $x_1,\ldots,x_n$ such that the balls of radius $\varepsilon$ around them cover all of $X$. Let $\alpha_0$ such that $\|S_\alpha x_j-x_j\|<\varepsilon$ for all $\alpha>\alpha_0$ and $j=1,\ldots,n$. Then for $y\in X$, there exists $j$ with $\|y-x_j\|<\varepsilon$ and, for $\alpha>\alpha_0$, $$ \|S_\alpha y-y\|\leq\|S_\alpha(y-x_j)\|+\|S_\alpha x_j - x_j\|+\|x_j-y\|\\ \leq K\|y_j-x\|+2\varepsilon<(2+K)\varepsilon, $$ where $K=\sup\|S_\alpha\|$.