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Both spaces are defined using the same set-builder principle: they are the collections of all measurable functions for which the corresponding norm (either the p-norm or the infinity norm) is finite.

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

Both spaces are defined using the same set-builder principle: they are the collections of all measurable functions for which the corresponding norm (either the p-norm or the infinity norm) is finite.

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

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

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

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

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

When p = +$\infty$, the infinity norm of a measurable function f is defined as the essential supremum of its absolute value over the set E.

Conditions: $p = +\infty$.; f is a measurable function on E.

When p = +$\infty$, the infinity norm of a measurable function f is defined as the essential supremum of its absolute value over the set E.

Conditions: $p = +\infty$.; f is a measurable function on E.

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

Conditions: The p-norm $\|f\|_p$ has been defined.; f is a measurable function.

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

Conditions: The p-norm $\|f\|_p$ has been defined.; f is a measurable function.

When $0 < p$ < +$\infty$, the p-norm of a measurable function f is defined as the $1/p$ power of the Lebesgue integral of the p-th power of its absolute value over the measurable set E.

Conditions: E is a measurable subset of $\mathbb{R}^n$.; f is a measurable function on E.; $0 < p < +\infty$.

When $0 < p$ < +$\infty$, the p-norm of a measurable function f is defined as the $1/p$ power of the Lebesgue integral of the p-th power of its absolute value over the measurable set E.

Conditions: E is a measurable subset of $\mathbb{R}^n$.; f is a measurable function on E.; $0 < p < +\infty$.