  The $H^m$-conforming virtual elements of any degree $k$ on any shape of
polytope in $\mathbb R^n$ with $m, n\geq1$ and $k\geq m$ are recursively
constructed by gluing conforming virtual elements on faces in a universal way.
For the lowest degree case $k=m$, the set of degrees of freedom only involves
function values and derivatives up to order $m-1$ at the vertices of the
polytope. The inverse inequality and several norm equivalences for the
$H^m$-conforming virtual elements are rigorously proved. The $H^m$-conforming
virtual elements are then applied to discretize a polyharmonic equation with a
lower order term. With the help of the interpolation error estimate and norm
equivalences, the optimal error estimates are derived for the $H^m$-conforming
virtual element method.
