  In this paper, we generalize the concept of unbounded norm (un) convergence:
let $X$ be a normed lattice and $Y$ a vector lattice such that $X$ is an order
dense ideal in $Y$; we say that a net $(y_\alpha)$ un-converges to $y$ in $Y$
with respect to $X$ if $\Bigl\lVert\lvert y_\alpha-y\rvert \wedge
x\Bigr\rVert\to 0$ for every $x\in X_+$. We extend several known results about
un-convergence and un-topology to this new setting. We consider the special
case when $Y$ is the universal completion of $X$. If $Y=L_0(\mu)$, the space of
all $\mu$-measurable functions, and $X$ is an order continuous Banach function
space in $Y$, then the un-convergence on $Y$ agrees with the convergence in
measure. If $X$ is atomic and order complete and $Y=\mathbb R^A$ then the
un-convergence on $Y$ agrees with the coordinate-wise convergence.
