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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 1900s, 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.

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Linear functionals don't really "come out of nowhere". Just as we can consider normal vectors to be column matrices (n x 1), we can do the same for "row matrices" (1 x n). Just as we can consider matrices as representing a linear transformation, the inner product arises naturally (in this same view) as $x^Ty$. What happens when we view the standard basis column vectors as row vectors? Then $e_j$ becomes the linear functional $\pi_j$, which projects a vector onto its j-th coordinate. –  David Wheeler Jul 2 '12 at 1:57
If I were exploring the difference between row vectors and column vectors, I would make two observations: row vector x column vector = scalar and column vector x row vector = matrix. That this would lead me to the concepts of linear functional or dual space seems unlikely to me. –  echoone Jul 2 '12 at 18:48
Dear echoone, This answer, while at an elementary level, may still shed some light. Regards, –  Matt E Jul 3 '12 at 13:11
The correspondence between linear transformations from an n-dimensional vector space to an m-dimensional vector space (over a field F) and the vector space of all mxn matrices with entries in F is often explictly or implicitly covered in most linear algebra courses. Much time is typically devoted to the linear transformation induced by an mxn matrix $A$, where $A$ acts on an nx1 column vector $x$ to produce an mx1 column vector $Ax$. Linear functionals (in this view) are just the special case $m = 1$. –  David Wheeler Jul 3 '12 at 13:16
@MattE I like that explanation. In fact, that is exactly how it was done historically according to the stuff I have read. –  echoone Jul 4 '12 at 21:02
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4 Answers

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!

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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.

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Nice. I will definitely have to read that book. –  echoone Jul 3 '12 at 13:10
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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.

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I disagree that the concept of a linear functional occurs "everywhere". The two examples you gave are well and good but they will not force me to start thinking about linear functionals in general. I am wondering what Riesz's motivation was in producing the RRT. You mention that the RRT can be used to define the cross product using determinants. I find that interesting. Do you have a reference? –  echoone Jul 2 '12 at 18:54
With regard to the definition of the cross product, see my answer to this post: math.stackexchange.com/questions/61526/… This characterization can be found, for instance, in Amann and Escher's Analysis II. Now, if you want to disagree with me that linear functionals are "everywhere", that's fine, but before you go down that road you might want to consider the fact that duality is a fundamental construct in functional analysis which itself is fundamental to quantum mechanics which is indeed, in a sense, everywhere. –  ItsNotObvious Jul 2 '12 at 20:44
Moreover, in the context of linear algebra in particular, one can provide a useful characterization of the transition matrix between two bases of a finite dimensional space in terms of some particular linear functionals. For this point of view, consult Algebra by Birkhoff/Mac Lane. If all these examples, together with example that David provided in a comment, are not sufficient to convince you that functionals/duality are worth considering, well...I'm not sure what else can be said. –  ItsNotObvious Jul 2 '12 at 20:59
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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.

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