The thing which is missing in the above material about metric spaces is any kind of algebra. In most applications, we are interested in adding things and multiplying things by scalars and so forth. This requires the notion of a vector space, also called a linear space. The simplest example is ℝn which is described next.
In this chapter, F will refer to either ℝ or ℂ. It doesn’t make any difference to the arguments which it is and so F is written to symbolize whichever you wish to think about. In multivariable calculus, the main example is where F = ℝ. However, it is nice to observe that things work more generally. As to notation, when it is desired to emphasize that certain quantities are vectors, bold face will often be used. This is not necessarily done consistently. Sometimes context is considered sufficient.