Kantorovich-Rubinstein norm and its application in the theory of Lipschitz spaces

We obtain necessary and sufficient conditions on a compact metric space ( K , ρ ) ( {K,ρ } ) that provide a natural isometric isomorphism between completion of the space of Borel measures on K K with the Kantorovich-Rubinstein norm and the space ( lip ⁡ ( K , ρ ) ) ∗ {( {lip( {K,ρ } )} )^*} or equivalently between the spaces Lip ⁡ ( K , ρ ) Lip( {K,ρ } ) and ( lip ⁡ ( K , ρ ) ) ∗ ∗ {( {lip( {K,ρ } )} )**} . Such metric spaces are studied and related properties of Lipschitz spaces are established.