A linear functional on a vector space V (over some field F, although you could take F=R usually) is a function φ:V->F which is linear:
  • ∀u,vV: φ(u+v) = φ(u)+φ(v)
  • ∀c∈F,v∈V: φ(cv) = cφ(v)

It turns out that the set of linear functionals over V, equipped with "natural" operations, is itself a vector space! See the dual of a vector space writeup for details on how to do this.

Functional analysis deals with many properties of lfs (linear functionals). In particular, while the lfs over a vector space of finite dimension are somewhat dull, lfs over function spaces and other spaces of infinite dimension are extremely "interesting".

As an example, note that every bounded sequence b in l(Z) defines a linear functional on the space of absolutely summable sequences l1(Z) by

φb(a) = (a1b1, a2b2,...,akbk,...)
Other functionals exist, but they are not continuous, so you usually don't see them in functional analysis.