Overview
Standard single particle Quantum Mechanics uses a Hilbert space that is essentially isomorphic to the space of complex valued functions. (with Dirac Delta functions as basis vectors).
That is, there is a Hilbert space of vectors, each representing the presence of a particle at each location in space. In the 1-d case, we have
{% \textit{H}_1 = |x_0 \rangle, |x_1 \rangle , ... %}
representing vectors in the Hilbert space for particles located at {% x_0, x_1,... %}
In addition, there is a Hilbert space representing two particle states.
{% \textit{H}_2 = |x_0,x_1 \rangle, |x_1,x_2 \rangle , ... %}
The hilbert space that Quantum Field theory includes the Hilbert spaces representing any number of particles.
{% \textit{H} = \textit{H}_1 \bigoplus \textit{H}_2 \bigoplus ... %}
That is to say
{% |x_0 \rangle \; \in \textit{H} \; and \; |x_0,x_1 \rangle \; \in \textit{H} %}
Lastly, there is inlcuded a vector which represents the vacuum with no particles in it.
{% |0 \rangle \; \in \textit{H} %}
Spin
For particles without spin, the Hilbert space given above is complete. For particles with spin, the spin value needs to be added to each vector.