Closed Unit Balls and Quotient Maps


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 V and a closed subspace M. Elementary considerations assert that the open unit ball of V is mapped onto the open unit ball V / M 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 V? It is easy to see that the projection map is a contraction, so its image is contained in the closed unit ball of V / M. Is the map onto in general? If not, what if we put some additional restrictions on V? In the above pdf we investigate this question which illustrates the usefulness of equipping V with different topologies.


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