OICC Press Mathematical Analysis and its Contemporary Applications 3 3 2021 06 21 A new version of the Hahn-Banach theorem in b-Banach spaces 10.30495/maca.2021.1929965.1011 EN Mohammad Reza Haddadi Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran. Hossein Alaeidizaji Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran. Vahid Parvaneh Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran. Journal Article 2021 06 21 In this paper, we introduce the notion of b-Banach spaces and we present some examples. Also, we give an important extension of the Hahn-Banach theorem in a b-Banach space with an application.