I never really bothered to ask this question and now it seems silly...but why do we always seem to define function spaces $X(U)$ (e.g. $L^2(U), BV(U)$ for open sets $U\subset\Bbb{R}^n$? What breaks if we consider closed $U$? For example, $L^2([0,1]^2)$ seems perfectly natural to me.
2026-03-31 14:22:36.1774966956
Why are function spaces typically defined on open sets?
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First, there is a difference between $C([0,1])$ and $C((0,1))$ - the spaces of functions that are continuous on the closed and open interval, respectively. There is no difference between $L^2([0,1])$ and $L^2((0,1))$ though.
If you define function spaces on open sets then every point of the set is an interior point, which is convenient if you want to work with differentiability.