# Essential Supremum vs. Uniform norm

I just went to check something about the $||\cdot||_\infty$ norm and realized that it can perhaps refer to two quite different things. I'm coming at this from an Analysis class so I am use to having $||f||_\infty = ess sup |f(t)|$. But, according to wikipedia (http://en.wikipedia.org/wiki/Supremum_norm) the uniform norm is the same notation but is defined as $||f||_\infty = sup |f(t)|$.

Unless I am really missing something, these are not equivalent. I'm a little uncomfortable with the possible ambiguity. How does one know which definition is actually being used in any one case? It seems like it is safe to say it is ess sup if anything explicitly references the $L^\infty$ space. I guesss I'm not sure what to ask. Can you verify that these are not necessarily equivalent? How do you differentiate which is being referred to?

The difference is in your underlying vector space. If its elements are functions (bounded, continuous, etc.), then it means $\sup$.
If its elements are equivalence classes (like in $L^\infty$), then you have to take ess-sup because there is a set of zero measure you have no control on.
• Actually, when it is about continuous functions in $L^\infty$, there is no other continuous function in its equivalence class. Hence, we are safe to think about the function even if it is its equivalence class. Aug 7, 2013 at 16:38