In this paper, we extend the notion of quotient of linear operators to linear relations. We introduce for two given linear relations [Formula: see text] and [Formula: see text], the linear relation quotient [Formula: see text] and we give a detailed treatment of some basic algebraic and topological properties of this new notion.

In this paper, almost closed subspaces and almost closed linear operators are described in a Hilbert space. We show Neubauer’s lemma and we give necessary and sufficient conditions for an almost closed operator to be with closed range and we exhibit sufficient conditions under which it is either closed or closable.

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This paper has triple main objectives. The first objective is an analysis ofsome auxiliary results on closedness and boundednes of linear relations. The seconde objective is to provide some new characterization results on semiclosed linear relations. Here it is shown that the class of semiclosed linear relations is invariant under finite and countable sums, products, and limits. We obtain some fundamental new results as well as a Kato Rellich Theorem for semiclosed linear relations and essentially interesting generalizations. The last objective concern semiclosed linear relation with closed range, where we have particularly established new characterizations of closable linear relation.