## Abstract

Hypernorm is a generalization of the notion of a norm on a vector space over a field. In this paper, we consider a hypervector space (V,+) over a hyperfield, where + is a hyperoperation, and prove that the hypernorm is continuous. We show that the natural linear transformation from V to VZ is continuous and open for all closed subhyperspaces Z of V. We prove BL(V,W), the set of all bounded linear transformations from V to W is a hyper-Banach space whenever W is complete. Furthermore, we obtain that in a hyper-Banach space V if {μ_{n}} is a sequence of continuous linear transformations with {/μ_{n}(u)/} is bounded for every u∈V, then {‖μ_{n}‖} is bounded. In the sequel, we prove several properties of hypernorm and linear transformations on hypernormed spaces.

Original language | English |
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Pages (from-to) | 1739-1756 |

Number of pages | 18 |

Journal | Journal of Analysis |

Volume | 32 |

Issue number | 3 |

DOIs | |

Publication status | Published - 06-2024 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology
- Applied Mathematics