How is the $L^p$ function space defined via the p-norm?

The L^p space is defined as the set of all measurable functions whose p-norm is strictly less than positive infinity.

The

L^p

space is defined as the set of all measurable functions whose p-norm is strictly less than positive infinity.

Conditions

Define the p-norm $\|f\|_p$ for a measurable function.
Evaluate the p-norm for various measurable functions.
Collect all measurable functions for which the calculated p-norm is finite (less than $+\infty$ ).
Formally denote this collection as the $L^p$ space.

The video writes the set-builder notation on the board: $L^p := \{ f \text{ measurable} : \|f\|_p < +\infty \}$ .

Assuming $L^p$ is inherently a Banach space from the start; the video notes it is initially just a set, and its linear space and completeness structures must be verified step by step later.
Confusing the space $L^p$ with the norm $\|f\|_p$ ; the norm is a value, while the space is the collection of functions for which that value is finite.

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