Banach was one of the founders of functional analysis. Spaces now called Banach spaces are more general than Hilbert spaces, in that the norm might not be derivable from a scalar product. In fact, a norm is so derivable if and only if it satisfies Appollonius's equality. Whereas Hilbert spaces have a simple classification (up to isomorphism) in terms of their dimension, the search for a classifying space for Banach spaces has been perhaps a waste of time.
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© by Ray Streater, 1/7/00.