Lp spaces

2020 Mathematics Subject Classification: Primary: 46E30 [MSN][ZBL]

In functional analysis, an $L^p$ space is a space of functions for which the $p$-th power of their absolute value is Lebesgue integrable. $L^p$ spaces are sometimes called Lebesgue spaces.

More precisely, let $(S, F, \mu)$ be a $\sigma$-finite measure space with $S$ the space, $F$ the $\sigma$-algebra of measurable sets and $\mu$ the measure. If the power is in the interval $1\le p \lt \infty$, then the $L^p$ space $L^p(S, F, \mu)$ contains the equivalence classes of complex measurable functions for which $$ \int_S |f(s)|^p \; d\mu(s) < \infty $$ where two functions $f$ and $g$ are equivalent if $f=g$ almost everywhere with respect to $\mu$.[1]

The $L^p$ norm of $f$ for $1\le p \lt \infty$ is $$ \| f \|_p = \left( \int_S |f(s)|^p \; d\mu(s) < \infty \right)^{1/p} $$

For $p = \infty$, the space $L^\infty(S, F, \mu)$ consists of all the equivalence classes of measurable functions on $S$ such that for a positive constant $M < \infty$, $$ |f(s)| < M $$ almost everywhere with respect to $\mu$.[1]

For $0\lt p \lt 1$, the $L^p$ norm does not satisfy the triangle inequality.[1]

References

[1] Stein, Elias M.; Shakarchi, Rami (2011). Functional Analysis: Introduction to Further Topics in Analysis. Chapter 1, Princeton University Press. ISBN 9780691113876.