Banach spaces $l_{p}\left( \mathbb{BC}\left( N\right) \right) $ with the $\ast -$ norm $\overset{..}{\parallel }.\overset{..}{\parallel }_{2,l_{p}\left( \mathbb{BC}\left( N\right) \right) }$ and some properties

Abstract

In this work, we construct vector spaces $l_{p}\left( \mathbb{BC}\left(N\right) \right) $ of absolutely $p-$ summable $\ast -$bicomplex sequences with the $\ast -$ norm $\overset{..}{\parallel }.\overset{..}{\parallel }_{2,l_{p}\left( \mathbb{BC}\left( N\right) \right) }$ over the field $\mathbb{C}\left( N\right).$ Also, we show that some inclusion relations hold and these vector spaces are Banach spaces by using Minkowski's inequality in $\mathbb{BC}\left( N\right) $ with respect to $\overset{..}{\parallel }.\overset{..}{\parallel }_{2}.$