Quantum Field Theory - Is Space Continuous?

In quantum field theory, the objects defined at each point are more complicated than vectors or numbers. At every single point there's an operator that acts on some hilbert space which itself is very hard to define. One way to get around it is just to pretend for these purposes that space isn't continuous. But there's a problem if we try to do that and i think it's one of the deepest problems in theoretical physics. There is a mathematical theorem that forbids you from writing down a discreet version of certain quantumfield theories. Among the class of quantum field theories that you cannot discreetise is the one that describes our universe - no kidding.