Can You Explicitly Construct a Hahn Banach Extension?

[StatusX]

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?

 

[StatusX]

The functional would have to have the following properties:

1. Any two sequences whose difference converges to zero must get the same value (eg, if f(1,0,1,0,...)=a then f(1/2,1/2,2/3,1/3,3/4,1/4,...)=a).

2. If the sum of two sequence converges to some limit, the sum of the values for each sequence equals the limit of their sum. (eg, if f(1,0,1,0,...)=a, then f(0,1,0,1,...)=1-a).

But what is, say, a? Can it be anything?