ABSTRACT
17.1 As we have seen in the preceding chapter, if E is infinite dimensional and U is an open subset of E, the compact-open topology τ0 on H(U;F) does not fully extend those properties which are normally valid when E is considered to be finite dimensional. Hence we are interested in finding a natural topology on H(U;F) which generalizes those essential topological properties of H(U;F) which were valid for several complex variables with respect to τ0. This natural topology τ, if there is one, must satisfy at least the following conditions:
τ ≥ τ0; τ = Tτ0 if E is finite dimensional.
https://www.w3.org/1998/Math/MathML"> d ^ m : H(U;F) → H U;P m E;F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065296/83108f8a-a772-47ac-b2d9-8e82b5d19535/content/ieq0339.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is continuous for τ.
τ-bounded subsets of H(U;F) are τ0-bounded sets.
(H(U;F),τ) is a barrelled space.
The Nachbin topology https://www.w3.org/1998/Math/MathML"> τ ω https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065296/83108f8a-a772-47ac-b2d9-8e82b5d19535/content/ieq0340.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> on H(0;F) which we study in this chapter satisfies the first three conditions easily and the fourth condition if E has an unconditional basis (see 18.16); in general, it is unknown that τω is a barrelled topology.