Consider the space of sequences of real numbers $\{\vec{a_i}\}$ where each $\vec{a_i}=\{a_{i_n}\}$ is such that $\sum_{n=0}^{\infty}2^n|a_{i_n}|<+\infty$. Then how could we better describe the space? Typically, does the space have countable linear basis,is it complete with respect to the norm $\|\vec{a_i}\|=\sum_n2^n|a_{i_n}|$,whether it is finite dimensional?
One example that comes to mind is of the sequence $a_n=\frac1{2^{2n}}$. Hence, I think the space is countable. How should I proceed further. Is any representation like Riesz representation work here? Thanks beforehand.
$\{(a_n):\sum 2^{n} |a_n| <\infty\}$ with $\|(a_n)\|= {\sum 2^{n} |a_n| }$ is separable infinite dimensional Banach space. In fact it just an $L^{1}(\mu)$ for a suitable measure $\mu$. [ Without absolute value you don't even get a norm as pointed out in the comments]. Define a measure $\mu$ on the power set of $\mathbb N$ by $\mu \{n\}=2^{n}$. Then $\int fd\mu =\sum 2^{n} f(n)$ for any $f:\mathbb N \to \mathbb R$ whcih is $\mu$ integrable. The identification of $f \in L^{1}(\mu)$ with the sequence $(f(n))$ shows that our space is nothing but $L^{1}(\mu)$. Note also that $(L^{1}(\mu))^{*}=L^{\infty}(\mu)=\ell ^{\infty}$.