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In the theory of Banach spaces, one defines the dual $E^*$ of a space $E$ to be the space of linear functionals $f:E\rightarrow \mathbb{R}$. In slightly more categorical language, in the category $\mathbf{Ban}$ whose objects are Banach spaces and whose arrows are continuous linear operators, we can define a functor $*:\mathbf{Ban}\rightarrow \mathbf{Ban}$ by $E\mapsto Hom(E,\mathbb{R})$. This is also has the nice side-effect that, for any continuous linear operator $T:E\rightarrow F$, $*(T):F^*\rightarrow E^*$ is the adjoint operator $T^*$ (and indeed, some more categorical treatments of functional analysis define the adjoint this way).

However, usually in category theory when one says that an object is the "dual", this refers to taking the dual of some universal mapping property: for example, the coproduct is the object which satisfies the dual UMP of the product.

My question is: in what sense, then, is $E^*$ the right dual object to $E$? Is it with respect to some mapping property in $\mathbf{Ban}$? Or is this the wrong perspective here?

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    $\begingroup$ "Dual" has many meanings in mathematics and there's no particular reason all of those meanings should be compatible. There is a very well-behaved notion of duality in symmetric monoidal categories which reproduces, for example, the dual of a f.d. vector space, and unfortunately it does not reproduce the Banach space dual. $\endgroup$ – Qiaochu Yuan Jun 26 '17 at 4:57
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So the answer was somewhere in the Nlab[1] after all! It turns out that closed categories are equipped with a notion of dual object (which they helpfully point out does not typically coincide with the dual object in the sense of monoidal categories, which Qiaochu mentioned) which generalizes the dual of a Banach space, once you recognize that $\mathbb{R}$ is the unit in $\mathbf{Ban}$.

[1]https://ncatlab.org/nlab/show/dual+object+in+a+closed+category

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