Let $\substack{\text{lim} \\ \leftarrow i} H\_i$ be a nuclear space, considered as the limit of the codirected diagram $$... \to H\_2 \to H\_1 \to H\_0,$$ 
with $f\_{ji}:H\_i \to H\_j$ being the trace class operators in the diagram. I'm hoping that $$(\substack{\text{lim} \\ \leftarrow i} H\_i)' \cong \substack{\text{colim} \\ i \rightarrow} H\_i$$ holds as topological vector spaces, so is this maybe a known relation? Here, the operators in the directed diagram are just the adjoints $f\_{ij}^\*:H\_i \to H\_j$.
Maybe there is some literature about the connection between projective and inductive limits of topological vector spaces?



Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)'$ and $\substack{\text{colim} \\ i \rightarrow } H_i$
The algebraic equality is clear by the remark of Paul Garrett. There is an issue about the usual locally convex topologies: The dual of $H\_\infty=\lim\limits\_{\leftarrow i}H\_i$ is usually endowed with the so-called strong topology of uniform convergence on bounded sets and the colimit has its colimit (or, as it is usually called, *inductive*) topology which is the finest locally convex topology making all maps $H\_i^\* \to H\_\infty^\*$ continuous. Spaces $H\_\infty$ for which these topologies coincide were called *distinguished* and Grothendieck proved
that e.g. reflexive Frechet spaces are distinguished. For this it would be enough to have a (projective) limit of arbitrary reflexive Banach spaces (neither Hilbert nor trace class connecting maps are needed). 


EDIT (concerning the algebraic equality).
That $H\_\infty$ is *topologically* the (projective) limit of the spectrum $(H\_i,f\_{j,i})$ means that $H\_\infty$ has the initial topology with respect to the canonical maps $f\_{j,\infty}:H\_\infty \to H\_j$. The $0$-neighborhood filter in $H\_\infty$ therefore has a basis $\lbrace f\_{j,\infty}^{-1}(V): V$ $0$-neighborhood in $H\_j\rbrace$. A continuous linear functional is bounded on some $0$-neighborhood and you can then well-define a continuous linear functional first on the range of $f\_{j,\infty}$ and then extend it by Hahn-Banach to $H\_j$. In this sense, every $\phi \in H\_\infty'$ comes from a continuous linear functional $\phi\_j$ on some $H\_j$, i.e., $\phi=\phi\_j \circ f\_{j,\infty}$.


