Let$U$be an open subset of a complex locally convex space$E$, and$H(U)$the space of holomorphic functions from$U$to$C$. If the dual$E′$of$E$is nuclear with respect to the topology generated by the absolutely convex compact subsets of$E$, then it is shown that$H(U)$endowed with the compact open topology is a nuclear space. In particular, if$E$is the strong dual of a Fréchet nuclear space, then$H(U)$is a Fréchet nuclear space.

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A new and fairly elementary proof is given of the result by B. Simon [S], that the potential in a Sturm-Liouville operator is determined by the asymptotics of the associated$m$-function near −∞. The proof given is based on relations between the classical transformation operators and the$m$-function.

We present an elementary and concrete description of the Hilbert scheme of points on the spectrum of fraction rings$k[X]$_{$U$}of the one-variable polynomial ring over a commutative ring$k$. Our description is based on the computation of the resultant of polynomials in$k[X]$. The present paper generalizes the results of Laksov-Skjelnes [7], where the Hilbert scheme on spectrum of the local ring of a point was described.