A note on weakly compact subgroups of locally quasi-convex groups

In this note two results about compact subgroups of locally quasi-convex groups are shown:

1. The quotient group of a locally quasi convex Hausdorff group modulo a compact subgroup is again locally quasi-convex.
2. If a subgroup of a locally quasi-convex group is compact in the weak topology, it is also compact in the original topology.