pointwise convergence of series? We have just been introduced to infinite series of functions, and immediately started working on uniform convergence and Weierstrass' M-test, and how the sum of the series behaves in terms of integration, differentiation, etc. 

But what about 'pointwise' convergence of a series of functions, like we had for sequences? What's the definition for that? Do I simply pick and $x$, and show that the series I get (which is one of numbers $a_n$) is convergent, or is there more to it?
Yes, it is the product topology. The trouble with this thing is that it is rarely metrizable and it's pretty nasty.
Yes, it is the product topology. The trouble with this thing is that it is rarely metrizable and it's pretty nasty.