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.

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.
• Here $T_{\lambda}$ is a net, two topologies are the same if they have the same converging nets. Dec 29, 2021 at 20:27
• Aahhh, I am a moron. Now, I understand why you picked $\lambda$ as an index. That is of course the way to go (+1). Dec 29, 2021 at 20:30