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

Source abstract

We obtain necessary and sufficient conditions on a compact metric space ( K , ρ ) \left ( {K,\rho } \right ) 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 ) ) ∗ {( {\operatorname {lip}( {K,\rho } )} )^*} or equivalently between the spaces Lip ⁡ ( K , ρ ) \operatorname {Lip}( {K,\rho } ) and ( lip ⁡ ( K , ρ ) ) ∗ ∗ {( {\operatorname {lip}( {K,\rho } )} )^{**}} . Such metric spaces are studied and related properties of Lipschitz spaces are established.