Convergence questions are treated by considering vector spaces V carrying a compatible topology, a structure that allows one to talk about elements being close to each other. Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if x and y in V , and a in F vary by a bounded amount, then so do x + y and a x . To make sense of specifying the amount a scalar changes, the field F also has to carry a topology in this context; a common choice is the reals or the complex numbers.