How is the $L^\infty$ space defined?

Conditions

• f is a measurable function.
• The infinity norm $\|f\|_\infty$ has been defined.

Reasoning, step by step

1. Define the infinity norm $\|f\|_\infty$ as the essential supremum.
2. Evaluate the infinity norm for various measurable functions.
3. Collect all measurable functions for which the infinity norm is finite.
4. Formally denote this collection as the $L^\infty$ space.

Example

The video writes the definition on the board: $L^{\infty}:=\{f\text{measurable}:\ \|f\|_{\infty}<+\infty\}$.

Common misconceptions

• Assuming $L^\infty$ consists of functions that are bounded everywhere; it consists of functions that are essentially bounded (bounded almost everywhere).
• Confusing the space $L^\infty$ with the set of all bounded functions in the classical sense.

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Explore next

• What common structural pattern do the definitions of the $L^p$ space and the $L^\infty$ space share?
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