In the Simon Reed text, after defining the strong operator topology it is said:
"A neighborhood basis at the origin is given by the sets of the form $\{S \ | \ S \in \mathcal{L}(X,Y), \|Sx_i\|_Y<\epsilon, \ \ i=1,...,n\}$ where $(x_i)_{i=1}^n$ is a finite collection of elements of $X$ and $\epsilon$ is positive."
I don't undestand why there must be a "finite collection $(x_i)_{i=1}^n$". Is it not enough to say "where $x$ is an element of $X$ and $\epsilon$ is positive"?
This is supposed to be a basis. If you only used a single element $x$, it would not be closed under intersections.