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.
Some infinite dimensional spaces, their elements and completion
-
0@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
2 Answers
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 $\mathbb R$.
To answer your question 1:
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.
-
0@user39729: Yes, the element $x_k=x_{(k_1,...,k_d)}$ is a real number, i.e. belongs to $\mathbb{R}$. – 2012-09-18