ABSTRACT
In this chapter, the author explores the idea of boundedness in multihyperseminormed, multihypernormed, multiseminormed and multinormed vector spaces, demonstrating a possibility of a variety of different explications, conceptualizations and formalizations of this idea. There is an essential difference between essential boundedness in semitopological vector spaces and essential boundedness in topological vector spaces. For instance, any compact set is essentially bounded in topological vector spaces but this is not true for semitopological vector spaces. Nevertheless, there are definite relations between essential boundedness and boundedness in hyperseminormed vector spaces.
