# ${L}^{p}$-space

## Definition

Let $(X,\U0001d505,\mu )$ be a measure space^{}. Let $$. The *${L}^{p}$-norm* of a function $f:X\to \u2102$ is defined as

$${||f||}_{p}:={\left({\int}_{X}{\left|f\right|}^{p}\mathit{d}\mu \right)}^{\frac{1}{p}}$$ | (1) |

when the integral exists. The set of functions with finite ${L}^{p}$-norm forms a vector space^{} $V$ with the usual pointwise addition and scalar
multiplication of functions. In particular, the set of functions with zero ${L}^{p}$-norm form a linear subspace of $V$, which for this article
will be called $K$. We are then interested in the quotient space $V/K$, which consists of complex functions on $X$ with finite ${L}^{p}$-norm,
identified up to equivalence almost everywhere. This quotient space is the complex ${L}^{p}$-space on $X$.

## Theorem

If $$, the vector space $V/K$ is complete^{} with respect to the ${L}^{p}$ norm.

## The space ${L}^{\mathrm{\infty}}$.

The space ${L}^{\mathrm{\infty}}$ is somewhat special, and may be defined without explicit reference to an integral. First, the ${L}^{\mathrm{\infty}}$-norm of $f$ is
defined to be the essential supremum^{} of $\left|f\right|$:

$${||f||}_{\mathrm{\infty}}:=\mathrm{ess}\mathrm{sup}\left|f\right|=inf\{a\in \mathbb{R}:\mu (\{x:\left|f(x)\right|>a\})=0\}$$ | (2) |

However, if $\mu $ is the trivial measure, then essential supremum of every measurable function^{}
is defined to be 0.

The definitions of $V$, $K$, and ${L}^{\mathrm{\infty}}$ then proceed as above, and again we have that ${L}^{\mathrm{\infty}}$ is complete. Functions in ${L}^{\mathrm{\infty}}$ are also called essentially bounded.

## Example

Let $X=[0,1]$ and $f(x)=\frac{1}{\sqrt{x}}$. Then $f\in {L}^{1}(X)$ but $f\notin {L}^{2}(X)$.

Title | ${L}^{p}$-space |
---|---|

Canonical name | Lpspace |

Date of creation | 2013-03-22 12:21:32 |

Last modified on | 2013-03-22 12:21:32 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 28 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 28B15 |

Synonym | ${L}^{p}$ space |

Synonym | essentially bounded function |

Related topic | MeasureSpace |

Related topic | Norm |

Related topic | EssentialSupremum |

Related topic | Measure |

Related topic | FeynmannPathIntegral |

Related topic | AmenableGroup |

Related topic | VectorPnorm |

Related topic | VectorNorm |

Related topic | SobolevInequality |

Related topic | L2SpacesAreHilbertSpaces |

Related topic | RieszFischerTheorem |

Related topic | BoundedLinearFunctionalsOnLpmu |

Related topic | ConvolutionsOfComplexFunctionsOnLocallyCompactG |

Defines | $p$-integrable function |

Defines | ${L}^{\mathrm{\infty}}$ |

Defines | essentially bounded |

Defines | ${L}^{p}$-norm |