ABSTRACT
This chapter teaches the properties of operators (mappings) between polyhyperseminormed and in particular, polyhypernormed, polyseminormed and polynormed vector spaces. Obtained results for hyperseminorms and hypernorms have diverse corollaries for norms and seminorms. The majority of these corollaries are new containing some classical results from functional analysis as special cases. The main emphasis is on linear operators and relations between their properties such as continuity and boundedness. The chapter recaps one of the basic concepts in mathematics – continuity. It defines continuity of functionals and hyperfunctionals. The statement for hypernorms is the direct corollary of the statement for hyperseminorms.
