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

Conditions

• The context is the summary of $L^p$ and $L^\infty$ definitions.
• Both $\|f\|_p$ and $\|f\|_\infty$ have been defined.

Reasoning, step by step

1. Identify the definition of the p-norm and the infinity norm.
2. Observe that $L^p$ is defined as the set of measurable functions with finite p-norm.
3. Observe that $L^\infty$ is defined as the set of measurable functions with finite infinity norm.
4. Conclude that both follow the pattern: Space := { f measurable : corresponding norm < +$\infty$ }.

Example

The video summarizes: 'both $L^p$ and $L^\infty$ are not newly introduced objects out of nowhere, but rather the collections of all measurable functions whose corresponding norms are finite.'

Common misconceptions

• Thinking the definitions are fundamentally different because one uses an integral and the other uses a supremum; the structural pattern of defining the space via the finiteness of the norm is identical.
• Assuming the spaces are defined by continuity or differentiability; they are defined solely by measurability and norm finiteness.

Watch the explanation

Explore next

• What is the definition of the p-norm for a measurable function when $0 < p < +\infty$?
• How is the $L^\infty$ space defined?