Two-norm convergence in the \(L_p\) spaces
The authors gave a representation theorem for two-norm continuous functionals defined on \(L_{p}\), \(1 \leq p \leq \infty\). A sequence \({f_{n}}\) of functions is said to be two-term convergent to \(f\) in \(L_{p}\), \(1 \leq p \leq \infty\) if there is an \(M>0\) such that \(\|f_{n}\|_{p}\leq M\) for all \(n\) and \(\lim_{n\rightarrow \infty} \int^{1}_{0} f_{n}(x)g(x)dx\) exists for every \(g\in L_{q}\) with \(\frac {1}{p}+\frac {1} {q} = 1\). A functional \(F\) defined on \(L_{p}\), \(1 \leq p \leq \infty\) is said to be two-norm continuous in \(L_{p}\) if \(F(f_{n})\rightarrow F(f)\) as \(n\rightarrow \infty\) whenever \({f_{n}}\) is two-norm convergent to \(f\) in \(L_{p}\). The authors proved that if \(F\) is a two-norm continuous linear functional on \(L_{p}\), \(1 \leq p \leq \infty\), then there exists a function \(g\in L_{q}\) with \(\frac {1}{p}+\frac {1} {q} = 1\), such that \(F(f)= \int^{1}_{0} fg\), for every \(f\in L_{p}\).
scientific article; zbMATH DE number 6032473