Unbounded norm topology beyond normed lattices

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.