Weak topology and the topology of pointwise convergence I'm reading the definition of weak topology in Banach Algebra Techniques in Operator Theory by Douglas:

According to an article about the product topology in Wikipedia, the product topology is also called topology of pointwise convergence.  I'm confused with the underscored sentence. According to the answer by @Brian M. Scott to the question What is the Topology of point-wise convergence?, I was expecting 

$
\lim_{\alpha\in A}f_\alpha=f
$
  if and only if $\lim_{\alpha\in A}f_\alpha(x)=f(x)$ for every $x\in X$ where $(f_\alpha)_{\alpha\in A}$ is a net in $\mathscr{F}$. 

Could anyone clarify what is going on? 

I do not know, what Douglas wanted to point out with this sentence, as pointwise convergence is usally used in the way, Brian explains in his answer, but you can, if you want, adjust things: 
Each $x \in X$ deinfes a function $\hat x$ on $\def\F{\mathscr F}\F$ by $\hat x\color \mathscr F \ni f \mapsto f(x)$. Now - by definition of $\mathscr T$ $x_\alpha \to x$ in $\mathscr T$ iff $\hat x_\alpha \to \hat x$. Thus $\mathscr T$ is (induced by) the topology of pointwise convergence on the subset $\{\hat x \mid x \in X\}$ of $Y^{\mathscr F}$.
