What is the definition of the $L^\infty$ norm when $p = +\infty$?

Conditions

• $p = +\infty$.
• f is a measurable function on E.

Reasoning, step by step

1. Identify the case where the exponent p equals positive infinity.
2. Instead of using an integral formula, switch to the concept of the essential supremum.
3. Define the infinity norm $\|f\|_\infty$ as the essential supremum of $|f(x)|$ over E.

Example

The video displays the formula on the board: $\|f\|_\infty := \operatorname*{ess\,sup}_{x \in E} |f(x)|$.

Common misconceptions

• Attempting to use the integral formula $\left( \int_E |f(x)|^p dx \right)^{1/p}$ for $p = +\infty$; the video explicitly states that the integral-type formula is no longer used for this case.
• Confusing the essential supremum with the ordinary pointwise supremum.

Watch the explanation

Explore next

• How is the essential supremum defined?
• What is the difference between an essential upper bound and an ordinary upper bound?