Suppose \(X\) is a topological space . The space of bounded continuous functions \(C^b(X)\) equipped with the uniform norm is Banach space .
Proof
The space \(C^b(X)\) is a closed subspace of \(l^\infty(X)\). Since \(l^\infty(X)\) is a Banach space
, (0x67c55fbe)
implies that \(C^b(X)\) is also Banach space.