2 corrected typo

I would suggest that the following three applications of category theory to functional analysis can be useful (they have points of contact with some of the earliest earlier answers): they concern three topics (which are related) and at the least provide a unifying thread---in my opinion, they do more---to many themes in abstract analysis (spaces of measures, distributions, analytic functionals, the Riesz representation theorem and Gelfand-Naimark duality). These are: extensions of categories, free topological vector spaces and extended duality.

Extensions of categories. The basic example is the extension of the category of Banach spaces (with continuous linear mapings as morphisms) to the class of locally convex spaces (Wiener) and convex bornological spaces (Buchwalter and Hogbe-Nlend). The first has, of course, long occupied a place in the mainstream of functional analysis, the latter less so. They can be regarded as the categories obtained by "adding inductive, respectively, projective limits. This informal notion has been formalised in the first edition of the book "Saks spaces and applications to functional analysis". Exactly the same process can be applied to the category of metric spaces (where we obtain that of uniform spaces), that of Banach algebras (convex bornological algebras and locally multiplicatively convex algebras). The examples can be multiplied indefinitely. We mention two further extensions which we shall refer to below---compactological spaces (Buchwalter) (inductive limits of compact spaces) and Saks spaces (adding projective limits to the category of Banach spaces with linear contractions as morphisms---paradoxically, despite the fact that this category {\it has} limits). These example shows that this process often leads to us rediscovering America. However, it does have the advantage over Columbus that we are rediscovering a plethora of continents with a single expedition, often ones which are known but have hardly been investigated so that a rescrutinising may be worthwhile (after all, Columbus himself had been anticipated by native Americans).

Free locally convex spaces. The basic example here is the following. If we start with the unit interval $I$, then we can consider the free vector space over this set. If we then supply it with the finest locally convex topology which agrees with the original one on $I$ and complete it, we obtain a locally convex space which has the universal property that every continuous function from $I$ into a Banach spaces lifts to a unique continuous linear operator thereon (and is characterised by this property). Not surprisingly, this is just the space of Radon measures on $I$. Our point is that this construction can be carried out in an infinity of analogous situations and leads to a unified aproach to a large array of spaces which are of interest in analysis. We mention a small sample---compact spaces (universal property for continuous functions), compactologies and completely regular spaces (bounded, continuous functions), metric spaces (bounded Lipschitz-continuous functions), uniform spaces (bounded uniformly continuous functions), compact rectangles in $n$-space ($C^\infty$-functions), compact manifolds (again $C^\infty$-functions), open subsets of $n$-dimensional complex space (holomorphic functions). These lead to a long list of interesting spaces (of bounded Radon measures, of uniform measures, of distributions, of analytic functionals) and many of their basic properties can be deduced from general principles which arise from this method of construction (most important example, duality theorems). Again, many of these spaces are known but the historical path to their discovery was long and stony. It is of advantage to have a natural unified approach to their construction. Again, there aare further important cases which are indeed known but seem to have passed out of the mainstream despite an obvious demand for them. A particularly important and, in my opinion, unfortunate example (and a perennial favourite for queries in this forum) is the topic of extensions of the Riesz representation, which we shall now discuss.

Extensions of duality. As we have just mentioned, the classical example is the Riesz representation theorem for compacta. It was initially shown by Buck (for locally compact space and later, by other researchers, to the class of completely regular spaces) that this can be extended to the non-compact case by using the so-called strict topology which can be most succinctly described here as the finest locally convex topology on the space $C^b(S)$ of bounded, continuous fuctions on $S$ which agrees with that of compact convergence on the unit ball for the supremum norm. Again a number of queries on this forum suggest that this topic which barely entered into the mainstream despite the prominence of its proponents (Buck, Beurling and Herz---mainly motivated by questions in harmonic analysis) and, sadly, seems to have vanished without a trace. There is a simple and natural scheme at work here. If we have a duality between two of the central catogories of analysis, then we can extend it in an obvious way to one between say the category obtiained by adding inductive limits to the first one and projective limits to the second one. This leads almost automatically to the above extension of the Riesz representation to the case of completely regular spaces (or, better, compactological spaces). If we take as our starting point one of the central dualities of abstract analysis (that between a Banach space and its dual as a Banach space, or, if one wants a symmetric duality, as a Waelbroeck space, between a compact space and the Banach space of continuous functions thereon, resp. the same space regarded as a $C^*$-algebra, between a metric space and the space of bounded, Lipschitz functioons, then we can apply this process to obtain a large classes of extended dualities which are useful in abstract analysis.

Why is this useful? Lack of space prevents an elaborate justification of these three methods but I would like to mention the following version of Occam's rasor. One can speculate that one of the reasons for the fact that many of these extended dualities failed to enter into the mainstream of abstract analysis lies in the fact that to analysists accustomed to the Banach space settings, the structures employed here seemed unattractively elaborate and artifical, not to say baroque. (The strict topology is not only not normable, but is not a member of the accepted "nice" classes of locally convex spaces---Frechet, $DF$-, even barrelled or bornological. Of course, any such extension must necessarily be more elaborate than the original duality and the above considerations show that the ones presented here are the simplest that can succeed in the given situations and so are inevitable.

1 [made Community Wiki]

I would suggest that the following three applications of category theory to functional analysis can be useful (they have points of contact with some of the earliest answers): they concern three topics (which are related) and at the least provide a unifying thread---in my opinion, they do more---to many themes in abstract analysis (spaces of measures, distributions, analytic functionals, the Riesz representation theorem and Gelfand-Naimark duality). These are: extensions of categories, free topological vector spaces and extended duality.

Extensions of categories. The basic example is the extension of the category of Banach spaces (with continuous linear mapings as morphisms) to the class of locally convex spaces (Wiener) and convex bornological spaces (Buchwalter and Hogbe-Nlend). The first has, of course, long occupied a place in the mainstream of functional analysis, the latter less so. They can be regarded as the categories obtained by "adding inductive, respectively, projective limits. This informal notion has been formalised in the first edition of the book "Saks spaces and applications to functional analysis". Exactly the same process can be applied to the category of metric spaces (where we obtain that of uniform spaces), that of Banach algebras (convex bornological algebras and locally multiplicatively convex algebras). The examples can be multiplied indefinitely. We mention two further extensions which we shall refer to below---compactological spaces (Buchwalter) (inductive limits of compact spaces) and Saks spaces (adding projective limits to the category of Banach spaces with linear contractions as morphisms---paradoxically, despite the fact that this category {\it has} limits). These example shows that this process often leads to us rediscovering America. However, it does have the advantage over Columbus that we are rediscovering a plethora of continents with a single expedition, often ones which are known but have hardly been investigated so that a rescrutinising may be worthwhile (after all, Columbus himself had been anticipated by native Americans).

Free locally convex spaces. The basic example here is the following. If we start with the unit interval $I$, then we can consider the free vector space over this set. If we then supply it with the finest locally convex topology which agrees with the original one on $I$ and complete it, we obtain a locally convex space which has the universal property that every continuous function from $I$ into a Banach spaces lifts to a unique continuous linear operator thereon (and is characterised by this property). Not surprisingly, this is just the space of Radon measures on $I$. Our point is that this construction can be carried out in an infinity of analogous situations and leads to a unified aproach to a large array of spaces which are of interest in analysis. We mention a small sample---compact spaces (universal property for continuous functions), compactologies and completely regular spaces (bounded, continuous functions), metric spaces (bounded Lipschitz-continuous functions), uniform spaces (bounded uniformly continuous functions), compact rectangles in $n$-space ($C^\infty$-functions), compact manifolds (again $C^\infty$-functions), open subsets of $n$-dimensional complex space (holomorphic functions). These lead to a long list of interesting spaces (of bounded Radon measures, of uniform measures, of distributions, of analytic functionals) and many of their basic properties can be deduced from general principles which arise from this method of construction (most important example, duality theorems). Again, many of these spaces are known but the historical path to their discovery was long and stony. It is of advantage to have a natural unified approach to their construction. Again, there aare further important cases which are indeed known but seem to have passed out of the mainstream despite an obvious demand for them. A particularly important and, in my opinion, unfortunate example (and a perennial favourite for queries in this forum) is the topic of extensions of the Riesz representation, which we shall now discuss.

Extensions of duality. As we have just mentioned, the classical example is the Riesz representation theorem for compacta. It was initially shown by Buck (for locally compact space and later, by other researchers, to the class of completely regular spaces) that this can be extended to the non-compact case by using the so-called strict topology which can be most succinctly described here as the finest locally convex topology on the space $C^b(S)$ of bounded, continuous fuctions on $S$ which agrees with that of compact convergence on the unit ball for the supremum norm. Again a number of queries on this forum suggest that this topic which barely entered into the mainstream despite the prominence of its proponents (Buck, Beurling and Herz---mainly motivated by questions in harmonic analysis) and, sadly, seems to have vanished without a trace. There is a simple and natural scheme at work here. If we have a duality between two of the central catogories of analysis, then we can extend it in an obvious way to one between say the category obtiained by adding inductive limits to the first one and projective limits to the second one. This leads almost automatically to the above extension of the Riesz representation to the case of completely regular spaces (or, better, compactological spaces). If we take as our starting point one of the central dualities of abstract analysis (that between a Banach space and its dual as a Banach space, or, if one wants a symmetric duality, as a Waelbroeck space, between a compact space and the Banach space of continuous functions thereon, resp. the same space regarded as a $C^*$-algebra, between a metric space and the space of bounded, Lipschitz functioons, then we can apply this process to obtain a large classes of extended dualities which are useful in abstract analysis.

Why is this useful? Lack of space prevents an elaborate justification of these three methods but I would like to mention the following version of Occam's rasor. One can speculate that one of the reasons for the fact that many of these extended dualities failed to enter into the mainstream of abstract analysis lies in the fact that to analysists accustomed to the Banach space settings, the structures employed here seemed unattractively elaborate and artifical, not to say baroque. (The strict topology is not only not normable, but is not a member of the accepted "nice" classes of locally convex spaces---Frechet, $DF$-, even barrelled or bornological. Of course, any such extension must necessarily be more elaborate than the original duality and the above considerations show that the ones presented here are the simplest that can succeed in the given situations and so are inevitable.