Uniform boundedness theorems for nearly additive mappings (Q1969666)

A map \(F\) from a group \((G,+)\) into a Banach space \(Y\) such that, for some constant \(K \geq 0\) and ``control'' functional \(\rho\) on \(G\), satisfies the inequality \[ \Biggl\|\sum_{i=1}^n F(x_i)-\sum_{j=1}^m F(y_j)\Biggr\|\leq K \Biggl\{\sum_{i=1}^n \rho(x_i)-\sum_{j=1}^m \rho(y_j)\Biggr\}, \tag{*} \] where \(x_i,y_j \in G\), and \(\sum_{i=1}^n x_i=\sum_{j=1}^m y_j\), is called ``zero-additive''. The smallest constant \(K\) for which (*) holds is denoted by \(Z(F)\). The main result is the following Theorem: Assume that, for fixed \(G\) and \(\rho\), for every zero-additive map \(F\) there exists an additive map \(A:G \to Y\) such that \[ \text{dist}(F,A)=\inf \{c\geq 0: \|F(x)-A(x)\|\leq c\rho(x),\;x \in G\;\}<\infty. \] Then \(\text{dist}(F,A)\leq KZ(F)\).