The K\"unneth formula for nuclear $DF$-spaces and Hochschild cohomology

We consider complexes $(\X, d)$ of nuclear Fr\'echet spaces and continuous boundary maps $d_n$ with closed ranges and prove that, up to topological isomorphism, $(H_{n}(\X, d))^*$ $\iso$ $H^{n}(\X^*,d^*),$ where $(H_{n}(\X,d))^*$ is the strong dual space of the homology group of $(\X,d)$ and $H^{n}(\X^*,d^*)$ is the cohomology group of the strong dual complex $(\X^*,d^*)$. We use this result to establish the existence of topological isomorphisms in the K\"{u}nneth formula for the cohomology of complete nuclear $DF$-complexes and in the K\"{u}nneth formula for continuous Hochschild cohomology of nuclear $\hat{\otimes}$-algebras which are Fr\'echet spaces or $DF$-spaces for which all boundary maps of the standard homology complexes have closed ranges. We describe explicitly continuous Hochschild and cyclic cohomology groups of certain tensor products of $\hat{\otimes}$-algebras which are Fr\'echet spaces or nuclear $DF$-spaces.