Nanostructured VS4 was in situ grown on defect-rich carbon nanofibers as a functional separator coating, which exerts the efficient entrapment and electrocatalysis of LiPS conversion.

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Abstract. This paper studies boundedness and closedness of linear relations, which include both single-valued and multi-valued linear operators. A new (single-valued) linear operator induced by a linear relation is introduced, and its relationships with other two important induced linear operators are established. Several characterizations for closedness, closability, bundedness, relative boundedness, and boundedness from below (above) of linear relations are given in terms of their induced linear operators. In particular, the closed graph theorem for linear relations in Banach spaces is completed, and stability of closedness of linear relations under bounded and relatively bounded perturbations is studied. The results obtained in the present paper generalize the corresponding results for (single-valued) linear operators to multi-valued linear operators, and some improve or relax certain assumptions of the related existing results.2010 AMS Classification: 47A06, 47B15, 47A55.

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