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Apr 6, 2021 at 14:48 comment added coudy Well, actually, the upper semi-continuous function is the first return in the open unit ball... which can be used instead.
Apr 6, 2021 at 14:25 comment added coudy @Wong The function $l$ is not continuous but it is upper semi-continuous, which is enough to conclude it is bounded. This argument is pretty close to Martin's.
Apr 6, 2021 at 0:29 comment added Willie Wong Is it obvious that $l$ is continuous? I tried to reason like this earlier but got stuck justifying this fact.
Apr 5, 2021 at 22:03 history answered David Hughes CC BY-SA 4.0