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This convergence has been investigated recently for $(X,p,E)=(X,\\lvert\\cdot \\rvert,X)$ under the name of $uo$-convergence, for $(X,p,E)=(X,\\lVert\\cdot\\rVert,{\\mathbb R})$ under the name of $un$-convergence, and also for $(X,p,{\\mathbb R}^{X^*})$, where $p(x)[f]:=|f|(|x|)$, under the name $uaw$-convergence. 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