Mathematically the definitions are as follows : if $H_n$ is a $n$-dimensional complex Hilbert space then its two different corresponding ``Fock space"(s) are often denoted as $F_{1}$ and $F_{-1}$ defined as, $F_1 = \oplus_{k=0}^{\infty} \mathrm{Sym}^k(H_n)$ and $F_{-1}= \oplus_{k=0}^{\infty} \Lambda^k(H_n)$.

Physically for a quantum field theory one "defines" its so-called Hilbert space as the dual of an implicit vector space over $\mathbb{C}$ whose basis is in bijective correspondence to the set of all possible values for all the classical fields that occur in the underlying Lagrangian.

Now my question is twofold,

Does this physical notion of a "Hilbert space of a QFT" correspond to the $H_n$ or some ``total Fock space" that can be defined from the first mathematical definition as, $\otimes_{i \in \text{Fields}} F^i_{p_i}$ where $p_i=$1 if the $i^{\text{th}}$ field is bosonic or $-1$ if it is fermionic? (..I guess this tensoring is needed because the QFT can have both fermionic as well as bosonic fields..)

If we agree as above that the states of a QFT live in such a "total Fock space" and not in the Hilbert space defined in the second paragraph then shouldn't the "quantum field" be mapping into a space of Hermitian operators on this total Fock space and not just the Hilbert space?