History of Dual Spaces and Linear Functionals

Question

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.

Answer 1

I believe theory developed in two stages here. The work of Frigyes Riesz and others in the early 1900's considered concrete examples, and they spoke about linear functionals without feeling any need to gather them into a structured set (dual space). An analogue is perhaps Weierstrass, who discussed the convergence of sequences of functions in the 1870's without using the notion of a function space with a norm or a topology.

The Riesz representation theorem is a good example of this. Riesz (1907) first defines what he means by a continuous linear operation on the space $L^2([a,b])$; this is, in slightly modernized notation, an operation which for any $f\in L^2$ gives a number $U(f)$ such that $U$ is a linear map and such that whenever $f_n\to f$ in $L^2$ we have $U(f_n)\to U(f)$. Then he shows that for each continuous linear operation $U$ there exists a function $k$ such that $U(f)=\int_a^b f(x)k(x)dx$ for all $f\in L^2([a,b])$.

Note by the way that the theory was developed in function spaces before finite-dimensional vector spaces. There were many examples of functionals on the form $f\mapsto \int f(x)g(x)dx$ well known at the time (cf. potential theory, or Cauchy's integral theorem), so representation theorems would look very nice.

It took another 20 years before abstract Hilbert spaces were defined, and when Riesz speaks of this theorem again in 1935, he can use an entirely modern notation: "For every continuous linear function $\ell(f)$ there is a unique representing element $g$ such that $\ell(f)=(f,g)$", where $(\cdot,\cdot)$ is the inner product on the Hilbert space.

The theory for Banach spaces progressed in a similar manner. First linear functionals on $C([a,b])$ and $L^p([a,b])$ were studied and representation theorems were found (ca. 1910). Then in the 1920's a more abstract theory was developed, and in Banach's monograph from 1932 the subject is fully mature with "spaces of type (B)" [Banach spaces] and "conjugate spaces" [dual spaces]. I guess it was necessary to have several similar-looking but different examples before it seemed worth while to construct a general theory.

By the way, no author is cited more often in Banach's monograph than Frigyes Riesz!

Answer 2

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.

Answer 3

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.

Answer 4

I can't really speak to the history, but the concept of a linear functional occurs everywhere in mathematics, not just in linear algebra, and thus it makes sense to consider the collection of all such functionals and thus arrive at the dual space. For instance, a couple of examples from analysis include:

1. If $X$ is any space with a complete norm the differential $df(p)$ at a point $p \in X$ of a differentiable function $f:X \rightarrow \mathbb{R}$ is a functional

2. If $C([a,b])$ denotes the space of all continuous real-valued functions defined on $[a,b]$ then the mapping $$ \int^b_a : C([a,b]) \rightarrow \mathbb{R} : f \mapsto \int^b_a f $$ is a functional by elementary properties of the integral

Now the Riesz theorem is very interesting because it basically says that if you are dealing with an inner product space that every linear functional can be expressed in terms of the inner product, and this fact has a wide range of applications. For instance, one can use the RRT theorem to define and guarantee the existence/uniqueness of the cross product of two vectors in terms of the determinant. It can also be used to define and guarantee the existence/uniqueness of the gradient of a function. So, the concept of duality arises in many contexts outside linear algebra. I find this pretty motivating.