quantum field theory - Fock space vs. wavefunctionals

There are at least two representations of the Hilbert spaces of quantum field theory. For a scalar field, we have

The Fock space representation, such that every state is represented as the Fock space of 'one particle' representations,

$$\mathcal H = \bigoplus_{n = 0}^\infty \text{Sym}(\mathcal H_1^{\otimes n})$$

with $\mathcal H_1^0 = \mathbb C$ and $\mathcal H_1 = \mathcal L^2(\Bbb R^3)$.

The wavefunctional representation, such that every state is represented by a functional on the space of configurations of the field on $\Bbb R^3$ which is square integrable with respect to some measure

$$\mathcal H = \mathcal L^2(\mathcal F[C(\Bbb R^3)], d\mu[\phi])$$

Since those two representations describe the same theory, I am guessing that there is some isomorphism between the two, but what would it be?