Studying linear algebra for my exam, I doubt arose. If I define the category $\mathcal{V}$ of all the vector spaces, and the functor $\mathcal{F}:\mathcal{V}\rightarrow \mathcal{V}$ given by $\mathcal{F}(A) = A^{*}$, where $A^{*}$ is the dual space of $A$, have any sense compose $\mathcal{F}\circ \mathcal{F}\circ ... \circ \mathcal{F}$ $n$ times? $\mathcal{F}$ have some particular propierty like differenciability or continuity? Sorry if the questions have some mistakes, I'm only an undergraduate student (a friend explain me that the "set" of all the vector spaces is a category, and the funtions between to categories are called functors).
Another question is, can I define a topology on $\mathcal{V}$, or a metric? If anyone can explain me or send me a link I'll really appreciate it.
If $V$ is finite-dimensional, then $V^*$ is isomorphic to $V$, but not naturally-so, unless you have an inner-product or a non-degenerate bilinear form defined on $V$. On the other hand, $V^{**}$ is naturally-isomorphic to $V$ by the pairing of v**:=v*(v), i.e., a functional on $V^{**}$ evaluates an element of v* at v . If $V$ is infinite-dimensional, then $V$ embeds into $V^{**}$ (into its continuous dul, actually; please see below) but the embedding is not always an isomorphism (if it is, you say V is reflexive.).
Note that if your vector space is normed, you have both a continuous dual and an algebraic dual. In the finite-dimensional case these two coincide, but they are always different in the infinite-dimensional case, by cardinality reasons alone; you can always construct non-continuous linear maps in these infinite-dimensional spaces.
I have never seen a way of deciding if this functor is continuous or differentiable; this is not done with functors, AFAIK, since categories do not often have a natural (or any at all) topology or norm. About the topology you can assign to $V^*$, there are many different topologies you can define on the dual :strong topology, weak topology, weak* topology, etc. You can maybe start with, e.g., http://en.wikipedia.org/wiki/Weak_topology and follow the links.