Communicated by Philip Hartman, March 29, 1971 1. Introduction. Current work on the extension of function theory to infinite-dimensional domains has led to the consideration of classes of analytic functions defined on Banach spaces, with Fréchet derivatives of a given type, e.g., nuclear, compact or integral. The existence theory of partial differential equations in this setting follows from [G] for the nuclear bounded case, and is given in [D] for formal power series of ce-/3-7-type. In this note we describe the duality theory (Theorem 1) and the existence theory (Theorem 2) of partial differential equations for a class of spaces of entire functions defined on a Hubert space, with Fréchet derivatives given by Hilbert-Schmidt operators. When the underlying Hilbert space is finite-dimensional, we recover results in [T, Chapter 9], in [B] and in [NS] (Fischer space). When the underlying space is a Hilbert space of squareintegrable functions, we obtain the wave functionals in the Fock representation of quantum field theory (cf. [NT]), subsuming some of the results proved independently in [R]. 2. Hilbert-Schmidt polynomials. Let E be a Hilbert space over the complex field C, with inner product (|), and E' the dual of E, with the inner product (u'\v')-(v|u) for u' = (\u), v' = (\v). Let £ /Vn = E'V * • * V-E' denote the w-fold symmetric product of E' [Gr, p. 191]. The Hilbert-Schmidt inner product on E' v n is characterized for decomposable elements by (u[ V * * • V U n I Vi V • • • V Vn) =-]£ (u vl \ v[) • • • {ll rn \ V n), n\ , the summation extended over all permutations ir of the indices. E$ n denotes the w-fold symmetric product equipped with the Hilbert
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