Gyro- L p spaces

In this paper we introduce a gyro-analogue of L p {Lᵖ} spaces; we achieve this goal by constructing normed gyrolinear structures on the set of functions from an arbitrary measure space to the Möbius gyrovector space ℳ s {Mₛ} and the Einstein gyrovector space ℰ s {Eₛ} . This gives rise to two major classes of gyro- L p {Lᵖ} spaces: Möbius s - L p {Lᵖ} gyrospaces and Einstein s - L p {Lᵖ} gyrospaces. These two spaces will form a normed gyrolinear space with respect to the ∥ ⋅ ∥ p ℳ s {\|·\|ₚ^{Mₛ}} gyronorm and the ∥ ⋅ ∥ p ℰ s {\|·\|ₚ^{Eₛ}} gyronorm, respectively. We show that the Einstein L p {Lᵖ} -gyrospaces and Möbius L p {Lᵖ} -gyrospaces are isomorphic, and the isomorphism between them also preserves the L p {Lᵖ} gyronorm, thus giving rise to a gyroisometry. Furthermore, we show that the Möbius and Einstein L p {Lᵖ}