##
**Equivalence of topologies and Borel fields for countably-Hilbert spaces.**
*(English)*
Zbl 1081.57019

The author examines the main topologies – weak, strong, and inductive – placed on the dual of a countably-normed space and the \(\sigma\)-fields generated by these topologies. In particular, he proves that for certain countably-Hilbert spaces the strong and inductive topologies coincide and the \(\sigma\)-fields generated by the weak, strong, and inductive topologies are equivalent.

The author motivates his study as follows: “Although these results have been used implicitely or explicitely in the literature, there appears to be no efficient or complete proof published or readily accessible without having to work through a vast literature. It is the purpose of this paper to present proofs of these results in a largely self-contained manner. For this reason we have included here proofs of various well-known results; these proofs serve the function of setting up arguments and notation for our principal objectives.”

The main results consist in the statement and the proof of the following theorems.

Theorem 4.16. Let \(V\) be a countably-Hilbert space with dual \(V'\). Then the inductive, strong, and Mackey topologies on \(V'\) are equivalent (i.e. \(\tau_s= \tau_i= \tau_m\)).

(The Mackey topology on a topological vector space \(E\) with dual \(E'\) is the topology of uniform convergence on all balanced convex weakly-compact subsets of \(E'\) [cf. Section 21.4 in G. Köthe, Topological vector spaces. I (Springer-Verlag, Berlin) (1969; Zbl 0179.17001)].)

Theorem 5.3. Let \(V'\) be the dual of a countably-Hilbert space \(V\) which has a countable dense subset. Suppose V’ to be endowed with a topology \(\tau\). If \(\tau\) is finer that \(\tau_w\) and the inclusion map \(i_n': V_n'\to V'\) is continuous for all \(n\), then the \(\sigma\)-fields generated by \(\tau\) and \(\tau_w\) are equal (i.e. \(\sigma(\tau_w)= \sigma(\tau)\)). In particular, the \(\sigma\)-fields generated by the inductive, strong, and weak topologies on \(V'\) are equivalent (i.e. \(\sigma(\tau_w)= \sigma(\tau_s)= \sigma(\tau_i)\)).

(The dual countably-normed space \(V\) is given by \(V'=\bigcup^\infty_{n= 1} V_n'\) with the inclusions \(V_1'\subset\cdots \subset V_n'\subset V_{n+1}'\subset\cdots\subset V'\), cf. Prop. 3.6 in the paper.)

The author motivates his study as follows: “Although these results have been used implicitely or explicitely in the literature, there appears to be no efficient or complete proof published or readily accessible without having to work through a vast literature. It is the purpose of this paper to present proofs of these results in a largely self-contained manner. For this reason we have included here proofs of various well-known results; these proofs serve the function of setting up arguments and notation for our principal objectives.”

The main results consist in the statement and the proof of the following theorems.

Theorem 4.16. Let \(V\) be a countably-Hilbert space with dual \(V'\). Then the inductive, strong, and Mackey topologies on \(V'\) are equivalent (i.e. \(\tau_s= \tau_i= \tau_m\)).

(The Mackey topology on a topological vector space \(E\) with dual \(E'\) is the topology of uniform convergence on all balanced convex weakly-compact subsets of \(E'\) [cf. Section 21.4 in G. Köthe, Topological vector spaces. I (Springer-Verlag, Berlin) (1969; Zbl 0179.17001)].)

Theorem 5.3. Let \(V'\) be the dual of a countably-Hilbert space \(V\) which has a countable dense subset. Suppose V’ to be endowed with a topology \(\tau\). If \(\tau\) is finer that \(\tau_w\) and the inclusion map \(i_n': V_n'\to V'\) is continuous for all \(n\), then the \(\sigma\)-fields generated by \(\tau\) and \(\tau_w\) are equal (i.e. \(\sigma(\tau_w)= \sigma(\tau)\)). In particular, the \(\sigma\)-fields generated by the inductive, strong, and weak topologies on \(V'\) are equivalent (i.e. \(\sigma(\tau_w)= \sigma(\tau_s)= \sigma(\tau_i)\)).

(The dual countably-normed space \(V\) is given by \(V'=\bigcup^\infty_{n= 1} V_n'\) with the inclusions \(V_1'\subset\cdots \subset V_n'\subset V_{n+1}'\subset\cdots\subset V'\), cf. Prop. 3.6 in the paper.)

Reviewer: Ioan Pop (Iaşi)

### MSC:

57N17 | Topology of topological vector spaces |

60H40 | White noise theory |

54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |

46A99 | Topological linear spaces and related structures |

### Citations:

Zbl 0179.17001
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\textit{J. J. Becnel}, Proc. Am. Math. Soc. 134, No. 2, 581--590 (2006; Zbl 1081.57019)

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### References:

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[2] | I. M. Gel\(^{\prime}\)fand and G. E. Shilov, Generalized functions. Vol. 2. Spaces of fundamental and generalized functions, Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer, Academic Press, New York-London, 1968. |

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[4] | Gottfried Köthe, Topological vector spaces. I, Translated from the German by D. J. H. Garling. Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York Inc., New York, 1969. · Zbl 0179.17001 |

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[8] | Yau-Chuen Wong, Introductory theory of topological vector spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 167, Marcel Dekker, Inc., New York, 1992. With a chapter by Mau-Hsiang Shih [Mou Hsiang Shih]. · Zbl 0759.46001 |

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