Later on, Lyra [2] proposed a further modification of Riemannian geometry and removed non-integrability of length transfer by introducing a gauge function into the structure-less manifold as a result of which a displacement vector arise naturally.
Serving as a generalization of the concept of d-summability, the authors of [5] proposed a novel modification by the use of a gauge function (dimension function) in order to use different functions of diameter.
The duality mapping J φ : E → E ∗ associated with a gauge function φ is defined by J φ ( x ) = { f ∗ ∈ E ∗ : 〈 x, f ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f ∗ ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E, where 〈 ⋅, ⋅ 〉 denotes the generalized duality pairing.
The duality mapping J φ : X ⟶ 2 X ∗ associated with a gauge function φ is defined by J φ ( x ) = { f ∗ ∈ X ∗ : 〈 x, f ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f ∗ ∥ = φ ( ∥ x ∥ ), ∀ x ∈ X }, where 〈 ⋅, ⋅ 〉 denotes the generalized duality paring.
Let E ∗ be the dual space of E. The duality mapping J φ : E → 2 E ∗ associated to a gauge function φ is defined by J φ ( x ) = { f ∗ ∈ E ∗ : 〈 x, f ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f ∗ ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E. In particular, the duality mapping with the gauge function φ ( t ) = t, denoted by J, is referred to as the normalized duality mapping.
Let E* be the dual space of E. The duality mapping J φ : E → 2 E * associated with a gauge function φ is defined by J φ ( x ) = { f * ∈ E * : 〈 x, f * 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ f * ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E. In particular, the duality mapping with the gauge function φ(t) = t, denoted by J, is referred to as the normalized duality mapping.
The duality mapping J φ : E → E ∗ associated with a gauge function φ is defined by J φ ( x ) = { x ∗ ∈ E ∗ : 〈 x, x ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ x ∗ ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E. In the case that φ ( t ) = t, J φ = J, where J is the normalized duality mapping.
