I'm trying to understand the Hahn Banach theorem, that every bounded linear functional f on some subspace M of a normed linear space X can be extended to a linear functional F on all of X with the same norm, and which agrees with f on M. But the proof is non-constructive, using zorn's lemma. So I'm trying to come up with examples so I can understand it better. For example, let L be the space of all bounded infinite sequences with the sup norm, and let M be the subspace of L consisting of those sequences that converge to some finite limit. Then, on this subspace, the functional given by taking the limit of a sequence is clearly linear and bounded. So it must extend to a bounded linear functional on all of L. But I can't imagine what such a functional would look like. Is it possible to explicitly construct one?