TITLE: History of Dual Spaces and Linear Functionals
QUESTION [17 upvotes]: Does anyone know or can anyone give a reference explaining how the concepts of a linear functional and particularly that of a dual space developed? I know Riesz published his famous representation theorem in the first decade of the $1900$s, but did he use these concepts then?
My main reason for asking this question is motivation: These two concepts seem to arise out of thin air in all linear algebra books that I have looked at. It would be nice to find a motivating (non-contrived) example that forces you to look at the dual space of a vector space.

REPLY [9 votes]: Dieudonne's History of functional analysis gives a very thorough account of the development of these notions.  
The theory of linear functionals and related ideas comes out of the theory of linear equations.  One paticular direction in which this was generalized, which was of particular importance for later developments, was that of the theory of integral equations, in part by thinking of an integral equation as a limit of a system of linear equations in an increasing number of unknowns.  Over time, this set of ideas developed into modern functional analysis.  See Dieudonne's book for (many) more details.

REPLY [5 votes]: I found two very interesting articles that explain the motivation behind many things including dual spaces and linear functionals. Here they are:
On the Hahn-Banach Theorem
The Hahn-Banach Theorem: The Life and Times
It's annoying that the history of these things is not discussed in most textbooks. A lot of things are making a lot more sense to me now.