  In a countably normed space which is a linear space equipped with a countable
number of pair-wise compatible norms, we prove the existence of a common
nearest point (in all norms) from a point outside a nonempty subset if this
subset is compact with respect to all norms. We also prove the uniqueness of
that common nearest point if the completion of the space equipped with only one
of its norms is uniformly convex.
