Generalizing algebraically defined norms (Q2147952)

Let \((\Omega,\Sigma,\mu)\) be a measure space and \(p:\Omega\to[1,\infty)\) be a measurable function. The variable Lebesgue space \(L^{p(\cdot)}(\Omega,\Sigma,\mu)\) consists of measurable functions \(f\) on \(\Omega\) for which there exists \(\lambda=\lambda(f)>0\) such that \(\varrho(f/\lambda):=\int_\Omega|f(x)/\lambda|^{p(x)}\,d\mu(x)<\infty\). It is a Banach space with respect to the norm \(\|f\|:=\inf\{\lambda>0:\varrho(f/\lambda)\le 1\}\). The main result of the paper says that if \(\|f\|=1\), then the following statements are equivalent: \begin{itemize} \item[(a)] the mapping \(\lambda\mapsto\varrho(f/\lambda)\) is continuous at \(\lambda=1\); \item[(b)] the function \(f\) belongs to the closure of the set \(\bigcup_{r<\infty}\{\chi_{\{p(\cdot)<r\}}f: f\in L^{p(\cdot)}(\Omega,\Sigma,\mu)\}\); \item[(c)] \(\varrho(f)=1\). \end{itemize}