Some infinite dimensional spaces, their elements and completion

I have been referred to a paper which uses following notations:
1. $\mathbb{Z}^d \subset \mathbb{R}^d$ : d-dimensional integer lettice;
2. $\mathcal{X}=\mathbb{R}^{\mathbb{Z}^d}$ : set of all sequences of the form $x=(x_k)_{k \in \mathbb{Z}^d}$;
3. $\mathcal{X}_0$ : set of all finite sequences in $\mathcal{X}$;
4. And concludes that the real Hilbert space $\mathcal{H} = l_2(\mathbb{Z}^d)$ is the completion of $\mathcal{X}_0$ w.r.t the norm generated by $(\phi,\psi) = \sum_{k \in \mathbb{Z}^d} \phi_k \psi_k, \quad \phi , \psi \in \mathcal{X}_0$.
My Questions:
1. In (2) above, how to visualize elements of this space (these sequences). I mean, elements of $\mathbb{Z}^d$ should be of the type $k=(k_1, \cdots, k_d)$ for $k_i \in \mathbb{Z}$. I cannot understand indexing of $x_k$ by a d-dimensional 'integer vector'. What type of elements are there in one such sequence?
2. How $\mathcal{H}$ completes $\mathcal{X}_0$?

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Do you know how to prove it in the case $d=1$? –  Vobo Sep 17 '12 at 21:50
What do you mean by 'finite sequence' in 3.? –  Berci Sep 17 '12 at 21:55
I am afraid I cannot prove it in the case $d=1$. I can understand 'finite sequence' when I know what is meant by (2). Let us put it this way: Please help me to differentiate between $\mathbb{R}^d$, $\mathbb{R}^{\mathbb{N}}$, $\mathbb{R}^{[0,1]}$,$\mathbb{R}^{\mathbb{Z}^d}$, $(\mathbb{R}^d)^{\mathbb{Z}^d}$. Or any reference where these structures are defined clearly. Many Many Thanks. –  user39729 Sep 18 '12 at 20:17
@user39729: For any two sets $X$ and $Y$, $X^Y = \{ f\colon Y\rightarrow X | f \text{ function}\}$. In case $X=\mathbb{R}$ and $Y=\mathbb{N}$, a real valued sequence $(a_n)_n$ of $X^Y$ is the function $a\colon\mathbb{N}\rightarrow\mathbb{R}$, $a(n)=a_n$. –  Vobo Sep 18 '12 at 20:47

Qu1: Since $\mathbb Z^d$ is countably infinite, for first glance you can consider $\mathbb N$ instead and simply speak about 'coordinates'

To question 2: completion of a metric space $M$ is, roughly speaking, the space of all (possibly not yet existent, fictive, ideal) limit points of all Cauchy sequences. Formally, one takes the set of all cauchy sequences and takes its quotient by $\bf a\sim\bf b \iff \lim(a_n-b_n) = 0$. For example, the completion of $\mathbb Q$ is $\mathbb R$.

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Indexing by a tuple of $d$ integers is basically equivalent to $d$ integer indices. Note that another way to write it would be as the set of functions $f:\mathbb Z^d\to\mathbb R$.
The elements of $x$ would have the form $x_{(k_1,\dots,k_d)}$. I wouldn't call it a sequence, though.
@user39729: As celtschk has stated, your $\mathcal{X}$ is the set of functions $f\colon \mathbb{Z}^d \rightarrow \mathbb{R}$. Like the set of real-valued sequences is the set of functions $f\colon \mathbb{N} \rightarrow \mathbb{R}$. –  Vobo Sep 18 '12 at 20:41
Sorry didn't get the point. For a fixed $k \in \mathbb{Z}^d$, the lement $x$ belongs to $\mathbb{R}$? –  user39729 Sep 18 '12 at 20:46
@user39729: Yes, the element $x_k=x_{(k_1,...,k_d)}$ is a real number, i.e. belongs to $\mathbb{R}$. –  Vobo Sep 18 '12 at 20:52