Abstract: | We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L
p
functions where translations are isometries, namely Marcinkiewicz spaces Mp{\mathcal{M}^{p}} and Stepanoff spaces Sp{\mathcal{S}^p}, 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M
p
) or even unbounded (Mp0{M^{p}_{0}}). We construct a function f that belongs to these spaces and has the property that all approximate identities fe * f{\phi_\varepsilon * f} converge to f pointwise but they never converge in norm. |