A Comparison of Two Generalisations of Triplets of Hilbert Spaces

TL;DR

This paper compares two generalizations of triplets of Hilbert spaces, analyzing their similarities, differences, and conditions under which they coincide or can be transformed into each other.

Contribution

It clarifies the relationship between closely embedded and Berezanskii's generalized triplets of Hilbert spaces, establishing conditions for their equivalence and conversion.

Findings

Identifies when the two triplet concepts coincide

Determines conditions for their differences

Provides methods to transform one into the other

Abstract

We compare the concept of triplet of closely embedded Hilbert spaces with that of generalised triplet of Hilbert spaces in the sense of Berezanskii by showing when they coincide, when they are different, and when starting from one of them one can naturally produce the other one that essentially or fully coincides.