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A simple example: If you are given the category of Hilbert spaces with the bounded linear mappings as morphism sets, then dualization is a contravariant endofunctor.

So we can talk about "qualitative" properties, or in other words, things which one might label as "soft analysis".

However, in contrast to this, "hard analysis" would not only ask for the dual morphism, but also would like to compare the norms of the morphism and the dual morphism. I have no clue whether category theoretical concepts are powerful enough to talk about such relations reasonably.

More generally, while I do not expect that estimates can be explicitly stated in these algebraic terms, I would like to express that many algebraic constructions are metrically well-behaved.

Suppose I am given some objects and morphism sets in the Hilbert space category, and build new objects and morphism sets from these, e.g. direct sums, tensor products, apply certain well-known functors. The morphism that are constructed are either with norm $1$ - e.g. inclusions and projections for the direct sum - or they are constructed through a functor, like dualization, such that their norms can be easily estimated in terms of the norms of their 'preimages'.

What does this tell me about the reach of category theory, and can we describe the metric behaviour of categorial constructions in categorial terms?

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    @Rasmus: We i$n$deed have this equivalence. But besides the algebraic statement, that the dualization functor is naturally equivalent to an endofunctor, I also would like to express in categorial terms that this equivalence holds. In contrast, the norm of the dual morphism could have no relation at all to the original morphism. Such a functor would be useless for analytic purposes.2011-11-11

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Forgive the brevity of this answer but I don't have enough time, I hope the following may help you.

It's indeed possible express lots of analysis in categorical term, for instance you can define a metric space as an enriched category over the category/linear order of real numbers, for more details you can take a look to this link. For more information about application of category theory in context of analysis John Baez web site is a good place where to start, there you can find a lot of material, links and references.

If I find some other time I'll add other stuff.

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    Sorry, thanks to have provided the corrections.2011-11-11
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You might be interested in learning the basics of Gel'fand duality, $C^*$-algebras (from a categorical viewpoint) (depending on what you know about CT you may find it boring or too difficult) and so on.