Abstract
In this chapter we generalize the results of Chapter 2 to infinite-dimensional Hilbert spaces. So let H be a Hilbert space and let B(H) be the set of all bounded operators on H. Here a notable point is that linear operators on finite-dimensional Hilbert spaces are automatically bounded, whereas in general they are not. Thus we impose boundedness as an extra requirement, beyond linearity. This is very convenient, because as in the finite-dimensional case, B(H) is a C*-algebra, cf. §C.1. At the same time, assuming boundedness involves no loss of generality whatsoever, since we can alway replace closed unbounded operators by bounded ones through the bounded transform, as explained in §B.21. Nonetheless, even the relatively easy setting of bounded operators leads to some technical complications we have to deal with.
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Landsman, K. (2017). Quantum physics on a general Hilbert space. In: Foundations of Quantum Theory. Fundamental Theories of Physics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-51777-3_4
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DOI: https://doi.org/10.1007/978-3-319-51777-3_4
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