Examples of incomplete normed barrelled spaces

Abstract

In this note we investigate spaces of the type $L_{\varepsilon}^{p}(\mu)=\lbrace f\in L^{p}(\mu);{\rm supp}f\in \varepsilon \rbrace$ where ε is an ideal of “small” measurable sets with certain properties. Typically, these spaces endowed with the p-norm are not complete and thus, classical Banach space theory cannot be used.However, we prove that for good ideals ε the normed space $L_\varepsilon ^{p}(\mu)$ is ultrabornological and hence barrelled and therefore many theorems of functional analysis like the closed graph theorem or the uniform boundedness principle are indeed applicable.