Characterizing the Fréchet spaces $E$ for which Grothendieck's germ space $P(\Omega_1,E)$ is complete, semi-reflexive, Montel, or distinguished

Background

All terminology from locally convex space theory is defined on first use. A Fréchet space is a complete metrizable locally convex topological vector space over $\mathbb{C}$. A barrel in such a space is a closed, absolutely convex, absorbing set; the space is barrelled (Grothendieck's French: tonnelé) if every barrel is a neighborhood of zero. The strong dual $F'$ is the continuous dual with the topology of uniform convergence on bounded sets; the bidual $F''$ is the strong dual of $F'$. $F$ is semi-reflexive if the canonical map $F \to F''$ is onto, and a Montel space (Grothendieck's class (M)) if it is barrelled and every bounded set is relatively compact; a Fréchet-Montel space is reflexive. A Fréchet space $F$ is distinguished - Grothendieck's definition in the 1953 paper: every weakly bounded subset of $F''$ lies in the weak closure of a bounded subset of $F$ - if and only if its strong dual $F'$ is barrelled; the paper notes that non-distinguished Fréchet spaces exist. A Fréchet space is quasinormable if its zero-neighborhoods can be approximated, in the sense of Grothendieck's factorization criterion, by bounded sets transferred from the Banach discs of the space; all Banach spaces are quasinormable, and this is the property covering the positive cases in the 1994 literature below.

A nuclear space is a locally convex space whose topology is generated by seminorms admitting Hilbert-Schmidt factorizations; a Schwartz space is a locally convex space in which every neighborhood of zero contains another neighborhood whose canonical transfer map between the associated normed spaces is compact - a strictly weaker property than nuclearity (every nuclear space is Schwartz). The completed projective tensor product $\widehat{\otimes}_\pi$ of two Fréchet spaces is the completion of the algebraic tensor product for the strongest locally convex topology making all continuous bilinear forms continuous. An LB-space (resp. LF-space) is a countable inductive limit of Banach (resp. Fréchet) spaces; such a limit is regular if every bounded subset of the limit is contained and bounded in one of the steps. A Silva space is a countable inductive limit of Banach spaces with compact linking maps; the scalar germ spaces $P(K)$, $K$ compact, are nuclear Silva spaces. Let $\Sigma$ be the Riemann sphere $\mathbb{C} \cup \{\infty\}$. For open $\Omega \subseteq \mathbb{C}^n$, $H(\Omega,E)$ is the Fréchet space of $E$-valued holomorphic functions with the compact-open topology (paper I also treats real-analytic analogues $\varepsilon(\Omega,E)$ on open subsets of $\mathbb{R}^n$). For arbitrary $\Omega_1 \subseteq \Sigma$, $P(\Omega_1,E)$ is Grothendieck's space of germs of $E$-valued holomorphic functions on $\Omega_1$ - functions holomorphic on some neighborhood of $\Omega_1$, vanishing at $\infty$ if $\infty \in \Omega_1$ - with the inductive-limit topology over neighborhoods of $\Omega_1$; for open $\Omega_1$ one has $P(\Omega_1,E) = H(\Omega_1,E)$. For compact $K \subset \mathbb{C}$ this is the space of germs $H(K,E) = \operatorname{ind}_n H^\infty(U_n,E)$ over a decreasing neighborhood basis $(U_n)$, where $H^\infty(U_n,E)$ is the space of bounded $E$-valued holomorphic functions on $U_n$ with the supremum seminorms; it is an LB-space when $E$ is Banach and an LF-space when $E$ is Fréchet. $\omega$ is the countable product of complex lines: a nuclear (hence Schwartz), non-normable Fréchet-Montel (hence semi-reflexive, reflexive, distinguished) space.

Prior results. In paper II, §7 (Theorem 9, pp. 88-89): for open $\Omega$ and Fréchet $E$, $\varepsilon(\Omega,E)$ (resp. $H(\Omega,E)$) is distinguished if and only if $E$ is; clause (b) gives the bounded-set/bidual refinement, and Theorems 5-8 with corollaries give the corresponding equivalences for weak compactness, semi-reflexivity, completeness and bidual identifications. The paper states immediately afterwards that 'theorems 5 to 9 can be stated abstractly for the topological tensor product of a nuclear space with an arbitrary (F)-space', and §7, Proposition 6 realizes $P(\Omega_1,E)$-type spaces as such topological tensor products. In §8 (pp. 90-95), for germ spaces with arbitrary $\Omega_1 \subseteq \Sigma$: Proposition 11 - $P(\Omega_1,E)$ is barrelled in each of two cases: (a) $E$ barrelled and $\Omega_2 = \Sigma \setminus \Omega_1$ closed; (b) $E$ a Fréchet space. Proposition 15 - if $E$ is the strong dual of a Fréchet space, or a closed subspace of such a dual (in particular a Banach space), then $P(\Omega_1,E)$ is complete. Proposition 16 - if $E$ is the strong dual of a Montel space (resp. of a reflexive Fréchet space), or a closed subspace thereof, then every bounded subset of $P(\Omega_1,E)$ is relatively compact (resp. $P(\Omega_1,E)$ is semi-reflexive). The closing example of §8 (pp. 94-95): for $E = \omega$ and $\Omega_1 = \{0\}$, $P(\{0\},\omega)$ is not semi-reflexive (although $\omega$ is Montel), is not complete, and its strong dual $P(O,\omega')$ ($O = \Sigma \setminus \{0\}$) is not barrelled - i.e. $P(\{0\},\omega)$ is not distinguished.

Later context. Bonet-Defant (1986) proved positive partial results on the question, considered by Grothendieck, whether the completed projective tensor product of two distinguished Fréchet spaces is again distinguished - the unrestricted two-Fréchet form of the Theorem 9 question. Taskinen (1988) answered it negatively, constructing a Fréchet-Montel space $F$ with $F \widehat{\otimes}_\pi F$ not Montel (and, as reported by later citing literature, not even distinguished); Taskinen (1989) gave further examples of non-distinguished Fréchet spaces; Bonet-Dierolf (1992) surveyed the resulting theory of distinguished Fréchet spaces (biduals, stability, characterizations for sequence-space and Moscatelli-type constructions). For the germ-space problem itself, Bonet-Domański-Mujica (1994) proved a linearization $H(K,X) \cong \mathcal{L}_b(G(K),X)$ for Banach $X$ (with $G(K)$ a canonical Fréchet space, $\mathcal{L}_b$ the continuous linear operators with the topology of uniform convergence on bounded sets), showed $H(K,X)$ is always a regular LB-space, and proved $H(K,X)$ complete whenever the Fréchet space containing $K$ is quasinormable or $X$ is complemented in its bidual; the paper explicitly records as open the questions whether every regular LB-space is complete (Grothendieck's problem, open since the 1950s) and whether $c_0(E_J) \cong E_J \widehat{\otimes}_\pi c_0$ is an LB-space for every Fréchet-Montel space $E$. Bonet-Domański (1998) developed tensor-product representations and characterizations for spaces of $E$-valued real-analytic functions and germs with Fréchet and LB range spaces. Wegner (2025, arXiv:2512.13161) records that the regular LB-space completeness problem 'has been open since the 1950s and originates in Grothendieck's early work', and gives homological evidence toward a positive answer without solving it.

Problem Statement

The problem left open by Grothendieck's 'Sur certains espaces de fonctions holomorphes II' (1953), §8: characterize, in terms of properties of the Fréchet space $E$ that do not reference $P(\Omega_1,E)$, those $E$ for which the germ space $P(\Omega_1,E)$ - for a fixed but arbitrary non-open $\Omega_1 \subseteq \Sigma$ (the Riemann sphere), the 1953 counterexample being the one-point germ space $P(\{0\},E)$; for open $\Omega_1$ the question is settled by §7, Theorems 5-9 - has each of the following properties: (i) completeness; (ii) semi-reflexivity; (iii) the Montel property (barrelledness itself holds for every Fréchet $E$ by Proposition 11(b), so the question is reflexivity/relative compactness of bounded sets); (iv) distinguishedness, i.e. barrelledness of the strong dual. For compact $K \subset \mathbb{C}$, these germ spaces are represented in the later literature as $H(K,E) = \operatorname{ind}_n H^\infty(U_n,E)$ over a decreasing neighborhood basis $(U_n)$. Where the admissible class of $E$ depends on the choice of $\Omega_1$ or $K$, determining that dependence is part of the characterization.

The 1953 paper fixes the boundary cases: every Banach space $E$, more generally every strong dual of a Fréchet space and every closed subspace of one, satisfies (i) (Proposition 15); strong duals of Montel and of reflexive Fréchet spaces satisfy (ii)/(iii) (Proposition 16); every Fréchet $E$ gives a barrelled $P(\Omega_1,E)$ (Proposition 11(b)); and $E = \omega$ - a Fréchet-Montel, hence distinguished and semi-reflexive, space - fails (i), (ii) and (iv).

References

1. A. Grothendieck, Sur certains espaces de fonctions holomorphes. I., Journal für die reine und angewandte Mathematik 192 (1953), 35-64. DOI 10.1515/crll.1953.192.35, https://doi.org/10.1515/crll.1953.192.35; https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/AG/AG-9.pdf
2. A. Grothendieck, Sur certains espaces de fonctions holomorphes. II., Journal für die reine und angewandte Mathematik 192 (1953), 77-95. DOI 10.1515/crll.1953.192.77, https://doi.org/10.1515/crll.1953.192.77; https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/AG/AG-10.pdf
3. J. Bonet, A. Defant, Projective tensor products of distinguished Fréchet spaces, Proceedings of the Royal Irish Academy, Section A 85 (1986), 193-199. Zbl 0575.46055, https://zbmath.org/3919113; JSTOR stable URL https://www.jstor.org/stable/20489224
4. J. Taskinen, The projective tensor product of Fréchet-Montel spaces, Studia Mathematica 91 (1988), 17-30. DOI 10.4064/sm-91-1-17-30, https://doi.org/10.4064/sm-91-1-17-30
5. J. Taskinen, Examples of non-distinguished Fréchet spaces, Annales Academiae Scientiarum Fennicae, Series A I Mathematica 14 (1989), 75-88. DOI 10.5186/aasfm.1989.1431, https://doi.org/10.5186/aasfm.1989.1431
6. J. Bonet, S. Dierolf, On distinguished Fréchet spaces, in: Progress in Functional Analysis, North-Holland Mathematics Studies 170, North-Holland, Amsterdam, 1992, pp. 201-214. ISBN 0-444-89378-4; Zbl 0785.46003, https://zbmath.org/88797
7. J. Bonet, P. Domański, J. Mujica, Complete Spaces of Vector-Valued Holomorphic Germs, Mathematica Scandinavica 75 (1994), 150-160. DOI 10.7146/math.scand.a-12510, https://doi.org/10.7146/math.scand.a-12510, https://journals.msp.org/mscand/article/view/1167/1166
8. J. Bonet, P. Domański, Real analytic curves in Fréchet spaces and their duals, Monatshefte für Mathematik 126 (1998), 13-36. DOI 10.1007/BF01312453, https://doi.org/10.1007/BF01312453
9. S.-A. Wegner, A homological approach to (Grothendieck's) completeness problem for regular LB-spaces, preprint (2025; v2 April 2026). arXiv:2512.13161, https://arxiv.org/abs/2512.13161