Above is a pdf about closed unit balls and quotient maps. This was the result of several discussions with Chris Gartland, a fellow graduate student.
Fix a Banach space and a closed subspace . Elementary considerations assert that the open unit ball of is mapped onto the open unit ball equipped with the usual norm (this should be compared to a general transformation where even in finite dimensions the image is an ellipse). What happens to the closed unit ball ? It is easy to see that the projection map is a contraction, so its image is contained in the closed unit ball of . Is the map onto in general? If not, what if we put some additional restrictions on ? In the above pdf we investigate this question which illustrates the usefulness of equipping with different topologies.