Let ( 𝑋 , 𝑑 ) be a compact metric space. In 1987, Bade, Curtis, and Dales obtained a sufficient condition for density of a subspace 𝑃 of little Lipschitz algebra l i p ( 𝑋 , 𝛼 ) in this algebra and in particular showed that L i p ( 𝑋 , 1 ) is dense in l i p ( 𝑋 , 𝛼 ) , whenever 0 < 𝛼 < 1 . Let 𝐾 be a compact subset of 𝑋 . We define new classes of Lipchitz algebras L i p ( 𝑋 , 𝐾 , 𝛼 ) for 𝛼 ∈ ( 0 , 1 ] and l i p ( 𝑋 , 𝐾 , 𝛼 ) for 𝛼 ∈ ( 0 , 1 ) , consisting of those continuous complex-valued functions 𝑓 on 𝑋 such that 𝑓 | 𝐾 ∈ L i p ( 𝐾 , 𝛼 ) and 𝑓 | 𝐾 ∈ l i p ( 𝐾 , 𝛼 ) , respectively. In this paper we obtain a sufficient condition for density of a linear subspace 𝑃 of extended little Lipschitz algebra l i p ( 𝑋 , 𝐾 , 𝛼 ) in this algebra and in particular show that L i p ( 𝑋 , 𝐾 , 1 ) is dense in l i p ( 𝑋 , 𝐾 , 𝛼 ) , whenever 0 < 𝛼 < 1 .
