$L^\infty$

Let \((X,\mcm,\mu)\) be a measure space. The space \(L^\infty(X)\) is \[ L^\infty(X) \coloneqq \theset{f\colon X\to \CC \suchthat f \text{ is measurable and essentially bounded}} = \theset{f\colon X\to \CC \suchthat f \text{ is measurable and } \norm{f}_\infty < \infty}, \] where \(\norm{f}_\infty\) is the essential supremum of \(\abs{f}\).