Let $M(X)$ be the space of all finite Borel $\mathbb{R}$-valued measures on $X$, where $X$ is metric space, and $m=M(\mathbb{N})$.
It is known that $$L^p[a;b]\subset L^q[a;b]\ \text{with}\ 1\leq q\leq p\leq \infty,$$ $$BV[a;b],C[a;b]\subset L^1[a;b],$$ $$C[a;b]\cap BV[a;b]\neq C[a;b], BV[a;b],$$ $$\text{there are natural isomorphism}\ BV[a;b]\,\widetilde{\to}\,M[a;b]\ \text{and inclusion}\ L^1[a;b]\hookrightarrow M[a;b],\ \text{etc.}$$
But for discrete analogues of these spaces we have (inclusions may not be continuous) $$\ell^q\subset \ell^p\ \text{with}\ 1\leq q\leq p\leq \infty,$$ $$\ell^1\subset bv\subset c,$$ $$\ell^1\neq bv\neq c,$$ $$\text{there are a natural isomorphisms}\ m\,\widetilde{\to}\,bv\,\widetilde{\to}\,\ell^1.$$ where $$bv=\big\{(x_n)_{n=1}^\infty\, \big|\, \sum\limits_{n=1}^\infty|x_{n+1}-x_n|<\infty\big\},$$ $$c=\big\{(x_n)\, \big|\, \exists\, \lim\limits_{n\to\infty}x_n\in\mathbb{R}\big\}.$$
So, if anyone knows examples in the spirit of the above, please share.
Remark 1. I know that it depends on definitions and that my question is rather inaccurate. However, I am interested in what can happen if, when moving to the discrete case, one of the main ideas underlying the definition of a space is preserved.
Remark 2. If we assume all sequences $x=(x_k)_{k\in\mathbb{Z}}$ are "smooth" and assume $x^\prime_k=(x_{k+1}-x_{k-1})/2$ then it is easy to define discrete analogues $d$ and $s$ of space of test functions $\mathcal{D}(\mathbb{R})$ and Schwartz space $\mathcal{S}_1$ respectively (the topology on them can be set following classical reasoning).
I have described the distributions on these spaces: if $\lambda\in d^\prime$ ($\lambda \in s^\prime$) then there is sequence $(\lambda_k)$ such that $$\lambda (x)=\sum\limits_{k=-\infty}^\infty \lambda_k x_k\ \text{(and}\ |\lambda_k|\leq C|k|^N\ \text{with some}\ C=C(\lambda)>0,\, N=N(\lambda)\in\mathbb{N} \text{).}$$
And for each sequence $(\lambda_k)$ functional $\lambda(x)=\sum \lambda_kx_k$ belongs to $d^\prime$ (if $|\lambda_k|\leq C|k|^N$ then $\lambda\in s^\prime$).