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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?