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I'm using finite sums, so Hamel bases. Have you ever encountered anything like what I describe: superimposing basis functions to create new basis functions that are still linearly independent but now also satisfy a new property?Sorry to keep hammering this, but when you use a Schauder basis this is an ordered basis ( because convergence is not always absolute) , so you may be changing the order of the terms of the sum? Are you doing finite sums ( Hamel bases, equality) or infinite sums ( Schauder bases, convergence).