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?

## 1 Answer

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., $\phi=\phi_j \circ f_{j,\infty}$.

duals, rather than just the $H_i$ themselves? If so, then this would be correct, because a continuous linear map to a normed TVS (such as the scalars) from a projective limit of Banach spaces factors through some limitand, for nearly definitional, easy reasons. Are $H_i$'s Banach? Hilbert? It would not be good to identify every Hilbert space with its dual. $\endgroup$