analysis question

Hi, I am studying multiresolution approximation and have the following
question: Suppose X is a Banach space with dual X', and that L is a
subset of X'. What are the conditions on X, X', L s.t. having l(x) for
all l\in L identifies exactly one x\in X? Any pointers to where
similar material is generally covered would also be of great help.
Thanks, -- Pouya

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