Let $E$ be finite or countably infinite.
Is there a simple notation to describe the (Hilbert) space of real, square-summable sequences indexed by $E$?
Effectively, I want the space to be $\mathbb{R}^{\textrm{card}E}$ equipped with the Euclidean inner product whenever $E$ is finite and $\ell^2$ whenever $E$ is countably infinite. However, I'm hoping there is a simpler notation for this notion.
A rather standard notation is one of the following: $L^2(E),$ $L^2(c_E)$ or $L^2(E, c_E),$ where $c_E$ is counting measure on $E$. Since $E$ is countable, the expected measure is the counting measure; that's why it might be enough with $L^2(E).$ Anyway, I suggest that you explain the notation shortly when you introduce it.