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,v∈V: 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 l^{1}(**Z**) by

φ_{b}(a) = (a_{1}b_{1}, a_{2}b_{2},...,a_{k}b_{k},...)

Other functionals exist, but they are not continuous, so you usually don't see them in functional analysis.