A proper velocity gyrovector space is a real inner product space V, with the proper velocity gyrogroup addition ⊕ ⊕ U and with scalar multiplication defined by r ⊗ ⊗ v = s sinh(r sinh (| v |/ s)) v /| v | where r is any real number, v ∈ V, v ≠ 0 and r ⊗ ⊗ 0 = 0 with the notation v ⊗ ⊗ r = r ⊗ ⊗ v.