In which mathematical space do the spinors act on? I'm studying QFT and from what I've learnt so far is that a general quantum field $\widehat{\phi}(x)$ can be decomposed (at least for the fermion case) as
$$\widehat{\phi}(x)=∫\frac{d^{3}p}{(2\pi)^{3/2}}\sqrt{\frac{m}{E_{p}}}\sum_{s}(\widehat{b}_{p,s}u(p,s)e^{-ip\cdot x}+\widehat{d}^{\dagger}_{pls}v(p,s)e^{ip\cdot x})\equiv\widehat{\phi}^{(+)}(x)+\widehat{\phi}^{(-)}(x)$$
Now, how should I interpret the multiplication of the spinor (let's say) $u(p,s)$ with the ladder operator $\widehat{b}_{p,s}$? Is it a tensor product? If yes, what are the vector spaces involved here? Ladder operators should live on the space of all the linear operators that act on the Fock space, but spinors where do they live? I'm trying to create some logical mathematical structure in my mind, this would help me also to understand the notion of adjointness for their product.
 A: Let $\psi$ be a spinor, like $u_s(p)$ and $v_s(p)$. In that case, $\psi$ is a $\mathbb{C}^4$ tuple with numeric entries. If $O$ is any operator in a Hilbert space the multiplication of $O$ by $\psi$ is meant to be a $\mathbb{C}^4$ tuple with operator entries where we just multiply each numeric entry of $\psi$ by the operator. More precisely if $\psi = (\psi_\alpha)$ then $\psi O =(\psi_\alpha O)$.
If you want to have a more precise description of that, then yes, the tensor product would be the correct construct. If ${\cal L}({\cal H})$ is the space of operators in the Hilbert space, an operator-valued spinor is an element of $\mathbb{C}^4\otimes {\cal L}({\cal H})$.
This is an instance of something more general. Imagine you have a vector space $V$ in which your field takes its values. In QFT it would be a representation space of the Lorentz group. Then the operator-valued field will be an element of $V\otimes{\cal L}({\cal H})$.
To connect to the component approach, let $\{e_i\}$ be a basis of $V$, like the canonical basis of $\mathbb{C}^4$. We know that any element of $V\otimes {\cal L}({\cal H})$ can be expanded in terms of decomposable tensors, so a generic $\Psi\in V\otimes{\cal L}({\cal H})$ must be of the form $$\Psi = \sum_\alpha v_\alpha \otimes O_\alpha.$$
Further decomposing $v_\alpha = \sum_i v_{\alpha}^i e_i$ we have that
$$\Psi = \sum_{\alpha,i} v_\alpha^i e_i\otimes O_\alpha = \sum_i e_i\otimes \left(\sum_\alpha v_\alpha^i O_\alpha\right)$$
This is analogous to an expansion of an element $\psi \in V$ in the basis, $\psi = \sum_i \psi^i e_i$, with the only difference being that the components are the operators $O^i = \sum_\alpha v^i_\alpha O_\alpha$. In summary, objects in $V\otimes {\cal L}({\cal H})$ take the form of an expansion in the basis of $V$ where the coefficients are operators, so that they can be viewed as "tuples of operators" as suggested in the first paragraph.
