Fock space and occupation number I have troubles to understand the concept of a Fock space.
We defined it as a direct sum of the 0-particle, single particle, two particle etc. Hilbert space.
Unfortunately, I am not sure if I understood this concept of the direct sum.
So this Fock space $F$ consists of  $F = \{(x_1,x_2,....) ; \text{what?}\}$? I mean what are the conditions on each component in this sequence?  
This seems to be related to the occupation number representation, but I don't quite see how:
We said that for bosonic states
$|n_1,n_2,... \rangle  = S_{+} |i_1,i_2,...,i_N \rangle \frac{1}{\sqrt{n1!n2!...}}$
I interpreted this as $n_i$ tells me how many particles are in state $i_i$, but since we only consider finitely many cases, I would conclude from this that $n_{m}$ for $m>N$ are zero, is this true?
EDIT: Although I got interesting answers about the Fock space, nobody so far has actually talked about this occupation number representation. You can also find this representation here at page 6: click
 A: First, the direct sum $S=V\oplus U$, of two vector spaces $V$ and $U$ is just the vector space constituted of "sums of vectors" $s=v+u:=(v,u)$ of each original vector space. The multiplication by scalar is viewed as obbeying the distributive property: $\alpha s=\alpha(v+u)=(\alpha v,\alpha u)$. The sum of two vectors of $S$ is just $s_1+s_2=(v_1+v_2,u_1+u_2).$
In the case of the Fock space, the idea is to have just one space with all possible combinations of states with different number of particles, including superposition of different numbers. So the direct sum is used to add up all the spaces $\mathcal{H_n}=\mathcal{H}\otimes\mathcal{H}\otimes\cdots\otimes\mathcal{H}$ of $n$ particles that is the $n$-fold tensor product of $\mathcal{H}$, the hilbert space of just one particle, with itself. We then define the Fock Space as
$$\mathcal{F}(\mathcal{H})=\bigoplus_{n=0}^\infty\mathcal{H_n}$$
where, $\mathcal{H}_0=\Bbb{C}$ corresponds to no particles. So, a general element of the $\mathcal{F}$ will be a sequence
$$\psi=(\psi_0,\psi_1,\psi_2,\dots),$$
where $\psi_n\in\mathcal{H}_n$ is an $n$ particle state. So a general state is the superposition of different states of different particle numbers. This discussion has not concerned the case of bosons and fermions statistics, that requires the space to be symmmetric or anti-symmetric. For this we just put positive or negatives permutations over each $\mathcal{H}_n$:
$$\mathcal{H}_n^\pm=\Pi^\pm\mathcal{H}_n$$
A: Maybe I could clarify the concept of a direct sum with a matrix example:
Assuming you have three matrices $A_0$, $A_1$, $A_2$, the direct sum of them is
$$A_0\oplus A_1\oplus A_2 = \left( 
\begin{array}{ccc}
A_0 & 0 & 0 \\
0 & A_1 & 0\\
0 & 0 & A_2
\end{array}
 \right)$$
You can see the resulting matrix (~ the Fock space) is a block-diagonal combination of the constituent matrices (~ Hilbert spaces).
Now, you can imagine that $A_0$ is acting on the 0-particle Hilbert space, $A_1$ acts on the 1-particle Hilbert space, and so on. In this toy model, a 0-particle state is represented by, say
$$\left| \psi_0 \right\rangle = \left(
\begin{array}{c}
1 \\
0\\
0
\end{array}
\right)
$$
and a 1-particle state is, say,
$$\left| \psi_1 \right\rangle = \left(
\begin{array}{c}
0 \\
1 \\
0
\end{array}
\right)
$$
A creation operator must move from $\left| \psi_0 \right\rangle$ to $\left| \psi_1 \right\rangle$, therefore it has to have off-diagonal elements, e.g.:
$$a^{\dagger} = \left( 
\begin{array}{ccc}
0 & 0 & 0 \\
\sqrt{1} & 0 & 0\\
0 & \sqrt{2} & 0
\end{array}
 \right)$$
Likewise the annihilation operator is
$$a = \left( 
\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & \sqrt{2}\\
0 & 0 & 0
\end{array}
 \right)$$
And the number operator
$$n = a^{\dagger}a = \left( 
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 2
\end{array}
 \right)$$
You can generalize this model to larger Fock spaces (to accommodate more than 2 particles) and build on it (ultimately leading to infinite-dimensional matrices, if you will), but it should give you a sense of what it means when people say Fock space is a direct sum of Hilbert spaces.
A: Considering direct sums, plainly, if you have one coordinate axes and you put another one on  it you made a 2-D space, and that is it...another way of looking at it is this: if you have some space V and two sub spaces s1 and s2, in this space V, AND if they dont cross, meaning no elements of one set are simultaneously in the other set, then V is direct sum of s1 and s2...in the same way, you combine hilbert spaces to form all possible combinations that can occur...
