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$$\langle \psi | \psi \rangle =

\langle x|x \rangle \cdot \langle y|y \rangle =

\langle x| \langle y| |x\rangle|y\rangle$$

I.e., the bra is ##\langle x| \langle y|##. But I swear I've also seen it defined as ##\langle y| \langle x|##, i.e. the order of the systems is switched. Am I crazy, or is there a use for this?

Edit: it seems I'm not the only one.