The appropriate space for the study of a system of identical bosons, for instance, is something like
\begin{equation} \tag{1} \mathbb{C}\oplus\mathcal{H}\oplus(\mathcal{H}\otimes\mathcal{H})_S\oplus(\mathcal{H}\otimes\mathcal{H}\otimes\mathcal{H})_S\oplus \ldots \end{equation}
where $\mathcal{H}$ is the one-particle Hilbert Space and the subscript $S$ denotes the symmetric sector of the products.
My question is: how is this different from
\begin{equation} \tag{2} (\mathcal{H}\otimes\mathcal{H}\otimes\mathcal{H}\otimes\mathcal{H}\ldots)_S~? \end{equation}
That is, intuitively my impression is that the first space 'converges' to this one.
For example: suppose a state of 3 bosons. Regarding (1), this state is specifically in the subspace
$$ (\mathcal{H}\otimes\mathcal{H}\otimes\mathcal{H})_S $$
but regarding (2), it is also in the sector that is 'non-vacuum' in 3 of the coordinates of the tensor product.
Why is (1) and not (2) the Fock Space?