In quantum mechanics, the Hilbert space is the space of complex-valued functions belonging to
In classical field theory, the configuration space of the field is an infinite-dimensional space. The single point denoted
In quantum field theory, it is expected that the Hilbert space is also the
Thus the intuitive expectation should be modified, and the concept of quantum configuration space should be introduced as a suitable enlargement of the classical configuration space so that an infinite dimensional measure, often a cylindrical measure, can be well defined on it.
In quantum field theory, the quantum configuration space, the domain of the wave functions
That physically interesting measures are concentrated on distributional fields is the reason why in quantum theory fields arise as operator-valued distributions.
The example of a scalar field can be found in the references