$\Phi$ is the set of functions $f: [a, b] \to \mathbb{R}$ where $f$ is bounded, non-decreasing and continuous on $[a, b]$. Is $\Phi$ compact with the norm $(,)=\sup|−|$?
2026-04-06 15:42:33.1775490153
Is the set of non-decreasing bounded continuous functions a compact set with the norm $(,)=\sup|−|$?
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Your set is unbounded and thus not compact (consider $f_n(x)=nx$).