The classical $L^p$ spaces are Banach spaces

The measure-theory page already proves the classical completeness theorem: The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$ builds the normed spaces $L^p(\mu )$, and Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$ proves they are complete. Therefore each $L^p(\mu )$ with $1\le p\le \mathrm{\infty }$ is a Banach space in the sense of Banach space.

The published $L^p$ completeness and the Banach-property wording records this as the agreement seam between the measure-theory track and the functional-analysis track. Nothing on the present page repeats the quotient construction or the Riesz-Fischer proof.

• The sequence spaces ${\mathrm{\ell }}^p$ are the counting-measure instances of that same theorem.
• Later pages may cite this remark for the Banach property without reopening the measure-theoretic development each time.