What is required in order to prove $||\cdot||_{\infty}$ is well defined (for ($L^\infty, ||\cdot||_{\infty}$)

Question

I've encountered an exercise that asks to prove that $||\cdot||_{\infty}$, defined on $L^\infty$, is well defined (the value $||f||_{\infty}$ is the infimum of the essential bounds of $f \in L^\infty$ ).

I believe this means I would have to show that this is indeed a norm (positivity, homogeneity and triangle inequality), but is there anything else? (in $L^1$, for example, there's the issue of representatives since the integrals of two functions in $L^1$ can be equal if the functions are equal up to a set of measure $0$, so one would have to show the norms of two such functions are the same).

Thanks!

Answer 1

No; that is to prove that $\|\cdot\|_\infty$ is a norm. The question here is: does the definition of $\|\cdot\|_\infty$ make sense? Of course, if $f$ is a function which has an essential bound, then $\|f\|_\infty$ is simply $\inf\{\text{essential bounds of }f\}$, and it is clear that this makes sense. But $L^\infty$ is not a set of functions; it is a set of equivalence classes of functions: $f\sim g$ if the set $\{x\mid f(x)\ne g(x)\}$ has Lebesgue measure $0$.

So, your goal is to prove that, if $f$ and $g$ are functions both of which have some essential bound, then $f\sim g\implies\|f\|_\infty=\|g\|_\infty$.

Answer 2

Since $L^\infty$ is a set of equivalence classes of functions (under the equivalence relation of almost everywhere equality) in $\mathscr{L}^\infty$ (which is usually defined as the set of essentially bounded functions), the only thing that could potentially stop it from being well defined would be if two functions $f,g\in\mathscr{L}^\infty$ generating the same equivalence class $[f]=[g]\in L^\infty$ would have a different value under $\lVert \cdot \rVert_\infty$ (as this is indeed well defined on $\mathscr{L}^\infty$ as the infimum is always well defined). Thus, to check that it is well defined, pick an element in $L^\infty$, and two functions $f,g\in\mathscr{L}^\infty$ representing this element, and show that $\lVert f\rVert_\infty = \lVert g\rVert_\infty$.