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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 $\mathcal{X}_0$?
Thanks in advance.

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    Do you know how to prove it in the case $d=1$?2012-09-17
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    What do you mean by '*finite* sequence' in 3.?2012-09-17
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    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.2012-09-18
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    @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 $a\colon\mathbb{N}\rightarrow\mathbb{R}$, $a(n)=a_n$.2012-09-18

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