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dEdt

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*subspaces*of a Hilbert space, rather than just two vectors in a Hilbert space?

The reason I'm asking is that in practice physicists often do exactly that. For example, in discussions about spin physicists will talk about, say, the amplitude for an electron that's spin up in the z direction to be spin up in the x direction. But [itex]|+z\rangle[/itex] and [itex]|+x\rangle[/itex] aren't vectors in the electron's Hilbert space (which includes position eigenstates tensored with spin eigenstates), they're subspaces of the Hilbert space. Yet there doesn't appear to be any problem with assigning an amplitude between them.

I can give numerous other examples of this. It really is commonplace. Yet I haven't seen any procedure for constructing an inner product between subspaces in any textbook.