How precisely are operator valued distributions defined?
Isn't it defined exactly the same as a real-valued or complex-valued distribution, except that it takes values in the set of linear operators on a Hilbert space? See also posts #34 and #35 in this thread (option 2 only, since you convinced me option 1 doesn't work, as acknowledged in #43).
Hm, maybe I should have said "the set of bounded linear operators"...not sure about that.
There's probably an exact definition in Streater & Wightman. That's the first place I'd look for one.
An operator valued distribution is a continuous linear map from the space of test functions to the space of operators over a Hilbert space. Quantum fields are operator valued distributions which have a Fourier transform.
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