This is sort of a reference request. I recall that there was a notation for functions that are $L^2$ on intervals $[n,n+1]$ for example and´these $L^2-$norms are uniformly bounded. It was something like $L^2,\text{unif}$ I think, but I do not recall details.
2026-04-29 11:52:10.1777463530
Uniformly $L^2$
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Maybe you're looking for locally $L^2$ functions, denoted by $L^2_{loc}$.
These are functions $f$ such that for every compact set $K$, $f \mathbb{1}_K \in L^2$.