Many of the aspects of QFT are traditionally done in ways incompatible with a rigorous mathematical treatment, calling for a variety of tricks to fix essentially what was caused by unjustified calculations or generalizations. One example for all is treating the Fock space as the state space of a collection of infinitely many harmonic oscillators, leading to infinite zero-point energy and related problems. This is at odds with the mathematical postulates of quantum physics, because

1. it is easily proven that Fock space over any separable single-particle state space is separable itself, which precludes the use of an infinite tensor product of $L^2(\mathbb{R})$ which is an inseparable space,
2. although the latter space can in principle be constructed, infinite eigenvalues don't make any kind of sense, so the sum of energies of all of the oscillators would not be assigned any self-adjoint operator on it.

I'm looking for recommendations on references that follow the Reed & Simon approach and build quantum field theory upon proper Hilbert space operator theory, evading such discrepancies from the beginning.

Many of the aspects of QFT are traditionally done in ways incompatible with a rigorous mathematical treatment, calling for a variety of tricks to fix essentially what was caused by unjustified calculations or generalizations. One example for all is treating the Fock space as the state space of a collection of infinitely many harmonic oscillators, leading to infinite zero-point energy and related problems. This is at odds with the mathematical postulates of quantum physics, because

1. it is easily proven that Fock space over any separable single-particle state space is separable itself, which precludes the use of an infinite tensor product of $L^2(\mathbb{R})$ which is an inseparable space,
2. although the latter space can in principle be constructed, infinite eigenvalues don't make any kind of sense, so the sum of energies of all of the oscillators would not be assigned any self-adjoint operator on it – it would simply not be an observable.

I'm looking for recommendations on references that follow the Reed & Simon approach and build quantum field theory upon proper Hilbert space operator theory, evading such discrepancies from the beginning. Practicality is secondary, I don't expect the reference to get me to any point where I could actually compute any real world problem.

Disclaimer: I haven't read all parts of R&S, it's possible some answer is also found therein.