Tom, Perhaps one of the most common usages of the Kronecker delta, and a usage we skirted in the discussion, is to establish biorthogonality between a vector space and its dual space. The Kronecker delta arises when given an indexed basis and its indexed dual set (which may or may not span the dual space), we take inner products of vectors in the first with vectors in the second. Because of linear independence in both sets, the inner product will take the value 1 when the vectors correspond and 0 otherwise. The Kronecker delta, in this case, is a manifestation of inner products and duality.
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