Vector spaces in second quantization Studying about fermionic commutation relations, the convention
I'm following is to consider a set of creation (destruction) operators
$\hat{a}_{i}^{\dagger}\left(\hat{a}_{i}\right)$ with $i=1,...,n$
acting on some Hilbert space $V$ where canonical commutation relations
are satisfied. Then it is said that the number operators $\hat{n}_{i}=\hat{a}_{i}^{\dagger}\hat{a}_{i}$
form a mutually commuting set of Hermitian matrices and thus there
exist $2^{n}$ states $\left|\psi\right\rangle $ which are simultaneous
eigenstates of all $n$ of these number operators, spanning a vector space $W$. These notes read that $W_{\perp}$ is the orthocomplement of $W$ in $V$.
My confusion is in distinguishing between $V$, $W$, and $W_{\perp}$.
Can you help me clear this out? A short answer is enough, maybe with
a very simple example QM system?
 A: The vector subspace $W$ will be spanned by states that can be labeled as follows:
\begin{align}
  |\vec k\rangle = |k_1, k_2, \dots, k_n\rangle
\end{align}
where $k_i = 0$ tells you that there is no fermion present in "mode" $i$ and $k_i = 1$ tells you that there is one fermion present in "mode" $i$.  There are $2^n$ such states because there are $2^n$ possible binary sequences of length $n$. These states are defined by acting an appropriate composite of creation operators on the ground state
$$
  |k_1, k_2, \dots, k_n\rangle = (a_1^\dagger)^{k_1}(a_2^\dagger)^{k_2}\cdots (a_n^\dagger)^{k_n}|0, 0, \dots, 0\rangle
$$
where the ground state is defined as the state that is annihilated by every one of the annihilation operators $a_i$.  It can be shown that the states $|\vec k\rangle$ are orthogonal; $\langle \vec k| \vec \ell\rangle = \delta_{\vec k, \vec \ell}$.  If $W$ is a subspace of Hilbert space $V$ having dimension larger than $W$, then $W_\perp$ will be the set of all states $|\psi\rangle\in V$ that are orthogonal to every state in $W$; $\langle \psi|\vec k\rangle = 0$ for all sequences $\vec k$.  This might happen if, for example, one considers a scenario in which the system actually has more than $n$ modes.  Say there are actually $N$ possible modes a given fermion can occupy, so in reality the states in $V$ would be written as
$$
  |k_1, k_2, \dots, k_n, k_{n+1}, \dots, k_N\rangle.
$$
Suppose, further, than one defines
$$
  |k_1, k_2, \dots, k_n, 0, \dots, 0\rangle = |k_1, k_2, \dots, k_n\rangle
$$
so that $W$ is a subspace spanned only by the states in which there are no fermions in the higher modes $n+1, n+2, \dots N$.  Then any state where there is a fermion in at least one of the higher modes will be in $W_\perp$.  As an example, the following state will be in $W_\perp$:
$$
  |\underbrace{0, 0, \dots, 0}_\text{first $n$ slots}, 1, 0, \dots, 0\rangle.
$$
