Liuyipei
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Kiefer.Wolfowitz 10:35, 28 February 2012 (UTC)
In infinite dimensions, an algebraic Hamel basis does not do much for analysis. A Schauder basis is more useful, but any Schauder basis (if one exists) has limitations for linear metric spaces.
Consider: A Banach space is a Hilbert space if and only if each of its closed subspaces has a closed complementary subspace (whose direct sum equals the whole space), a result of Lindenstrauss and Tzarafi, I believe, from the 1960s.