What would be the "action" in functional analysis? I am reading Simmons' "Topology and Modern Analysis". He keeps bringing up the idea of studying the set of all structure preserving mappings to obtain info regarding the structure of a certain normed linear space, referring to representation theory of group and rings. The concept of an action is something I am somewhat familiar with in this kind of situation, however I fail to see the resemblance.
Relevant extract from the aforementionned book:

One of the basic principles of strategy in the study of an abstract 
  mathematical system can be stated as follows: consider the set of all 
  structure-preserving mappings of that system into the simplest system 
  of the same type. This principle is richly fruitful in the structure theory 
  (or representation theory) of groups, rings, and algebras, and we shall see 
  in the next section how it works for normed linear spaces. 
  We have remarked that the spaces $\mathbb{R}$ and $\mathbb{C}$ are the simplest of all normed linear spaces. If $N$ is an arbitrary normed linear space, the above principle leads us to form the set of all continuous linear transformations of $N$ into $\mathbb{R}$ or $\mathbb{C}$, according as $N$ is real or complex. 

 A: Since nobody dares, I shall try to break the ice. Keep in mind that the question bears the tag "soft-question", so accordingly this will be a "soft answer".
Answering your question will be an axercise into understanding what is really meant by "dual" in this context. It would be a mistake to think that the word is used in the usual, rigorous way: think of some Hilbert space; its topological linear dual will be isomorphic to it, therefore studying it cannot bring anything new that wasn't already contained in the original space. Therefore, "dual" means something different here.


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*Think of a topological space: what is its dual? Mathematically, it is not defined, but objects dual to a space should take points in that space as arguments and return numbers. Therefore, a good candidate should be the algebra of continuous functions on that space. Indeed, the Gelfand-Naimark isomorphism shows that there is a nice (natural) correspondance between compact Hausdorff topological spaces and commutative $\Bbb C *$-algebras with unity: all the topological properties of your space admit a $\Bbb C *$-algebraic translation (a thing massively speculated by Connes in his construction of non-commutative geometry). You can view such a "dictionary" in pages 6 and 13 of this document.

*A related yet a totally different approach is to associate to every topological space its structure sheaf. Again, plenty of topological and geometrical features of the space get a purely algebraic description by moving to its algebraic "dual". Since this is usually not considered a topic in functional analysis I shall not insist on it (this is used more in algebraic geometry done with schemes, the underlying topology being the Zariski topology).

*A similar thing can be done not for topological spaces, but for probability spaces: their duals will now be $W *$-algebras. Terrence Tao has a long but interesting post about this on his blog.

*A final example comes from a somewhat different direction: topological groups. Remember that the Gelfand-Naimark isomorphism mentioned above is constructed using the concept of "character" on a $\Bbb C *$-algebra. Similarly, the concept of Pontryagin dual of a locally-compact commutative topological group is constructed using the concept of "character" in the category of topological groups. The Pontryagin dual manages to grasp a lot of properties of the original group; in fact, it grasps all of them such that the dual of the dual is the original group again. Properties of the groups can be translated into properties of the dual (which is itself a locally-compact commutative topological group - but not necessarily isomorphic to the original one) and, more importantly, various spaces of functions on these groups can be put in correspondence with each other (not necessarily isomorphic but quite close) through the Fourier transform (in fact, the group algebras of the group and of its dual are isomorphic). Again, Terrence Tao has a very detailed post with what a working mathematician is supposed to know on the subject.
Now, you might ask whether all this is useful. Well, for one it creates bridges between various chapters of mathematics that were previously thought to be unrelated. Second, these constructions transform topological properties of a space into algebraic properties of its dual (with the exception of the Pontryagin duality which takes groups into groups), showing that many of the "visual" properties of some objects can be translated into purely "intellectual" properties about their duals. Third, the previous fact allows to extend some concepts by algebraically generalizing their duals: for instance, you would have a hard time coming with a concept more general than the one of topological space; and yet, translating things in terms of $\Bbb C *$-algebras, you may now speak of "non-commutative" spaces, a thing previously impossible in purely topological terms.
The Pontryagin duality is a duality of a different nature: instead of taking objects from one category (topological spaces) to objects of some other category (algebras), it stays inside the same category (topological groups). Its practical advantage is that to a potentially complicated group it may associate a simpler dual which could be easier to study. Results obtained for the dual might then be brought back onto the original group by taking the dual once more.
