Gyro- L ^p spaces

Abstract

In this paper we introduce a gyro-analogue of

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 {\mathcal{M}_{s}}

and the Einstein gyrovector space

ℰ s {\mathcal{E}_{s}}

. This gives rise to two major classes of gyro-

L p {L^{p}}

spaces: Möbius s -

L p {L^{p}}

gyrospaces and Einstein s -

L p {L^{p}}

gyrospaces. These two spaces will form a normed gyrolinear space with respect to the

∥ ⋅ ∥ p ℳ s {\|\cdot\|_{p}^{\mathcal{M}_{s}}}

gyronorm and the

∥ ⋅ ∥ p ℰ s {\|\cdot\|_{p}^{\mathcal{E}_{s}}}

gyronorm, respectively. We show that the Einstein

L p {L^{p}}

-gyrospaces and Möbius

L p {L^{p}}

-gyrospaces are isomorphic, and the isomorphism between them also preserves the

L p {L^{p}}

gyronorm, thus giving rise to a gyroisometry. Furthermore, we show that the Möbius and Einstein

L p {L^{p}}

gyronorms induce a complete metric space structure on the Möbius

L p {L^{p}}

gyrospace and the Einstein

L p {L^{p}}

gyrospace, respectively.