Equivalent definitions for Strong Operator Topology in Banach Spaces

Strong Operator Topology (definition-$1$): We called a sequence $\{T_n\}$ in $\beta(X,Y),$ for $X,~Y$ are Banach spaces and $\beta(X,Y)$ denotes the family of bounded linear operators from $X$ to $Y$, converges strongly to $T$ if for all $x$ in $X$, $T_nx$ converges to $Tx$. That is, we can say $$\lim_{n \to \infty}\|T_nx-Tx\|=0,~~\text{ for each }~x \in X.$$ Now the associated topology is known as Strong Operator Topology.

Strong Operator Topology (definition-$2$): Let us consider a family of maps $\{F_x:\beta(X,Y) \to Y \text{ such that } F_x(T)=Tx,~T \in \beta(X,Y)\}$ where $x\in X.$ Then Strong Operator Topology is defined as the weak topology generated by the family of maps $F_x$ for each $x \in X$.

I am not able understand the equivalence of definition $1$ and $2$. How can I prove both the definitions are same (equivalent)? I know the definition of weak topology in Banach spaces. But I am not able to understand the equivalence of aforesaid definitions. Can you please help me to understand these definitions. Thank you for your time.

Solution 1:

We want to show that a net $T_{\lambda}$ converges to $T$ with respect to $\textbf{def}$ 1 if and only if a net $T_{\lambda}$ converges to $T$ with respect to $\textbf{def}$ 2.

Suppose that $T_{\lambda}\to T$ with respect to $\textbf{def}$ 1, then we know that $T_{\lambda}x\to Tx$ for each $x\in X$ meaning that $$F_x(T_{\lambda})=T_{\lambda}x\to Tx=F_x(T).$$ Putting this in context, for each $F_x\in \{F_x:\beta(X,Y)\to Y \text{ such that }F_x(T)=Tx, T\in\beta(X,Y)\}$ we have $F_x(T_{\lambda})\to F_x(T)$. So $T_{\lambda}\to T$ with respect to $\textbf{def}$ 2.

On the other hand, if $T_{\lambda}\to T$ with respect to $\textbf{def}$ 2, then we know that $$F_x(T_{\lambda})=T_{\lambda}x\to Tx=F_x(T),$$ for each $x\in X$ and as a result $T_{\lambda}x\to Tx$ for each $x\in X$ and we have convergence with $\textbf{def}$ 1. It is simply unpacking the definitions.