You need the function to give a result that has the same properties, for some specified set of properties, regardless of the basis. GP's question is perfectly

You need the function to give a result that has the same properties, for some specified set of properties, regardless of the basis. GP's question is perfectly reasonable if you haven't already accepted that this set of properties is the only set of properties that matters. The thesis could be better stated "if we define multiplication to have properties X, you can't multiply vectors" which is much less divisive. I think GP's (and my question) is why those properties and none others are important.