Lp Spaces

The Lp spaces collect functions whose p-th power is integrable, forming complete normed spaces that are the bridge between measure theory and functional analysis.

Definition

For a measure space and an exponent p at least one, the space Lp consists of equivalence classes of measurable functions whose absolute value raised to the power p has a finite integral, normed by the p-th root of that integral.

Scope

This topic covers the Lp norm and the identification of functions equal almost everywhere, Holder's and Minkowski's inequalities, the completeness of Lp expressed by the Riesz-Fischer theorem, the special Hilbert-space case of square-integrable functions, duality between conjugate exponents, and density of simple and continuous functions.

Core questions

• Why must elements of Lp be equivalence classes of functions rather than functions?
• Which inequalities make the Lp norm a genuine norm and control products of functions?
• Why is each Lp space complete, and why does that matter?
• How are the duals of the Lp spaces identified through conjugate exponents?

Key theories

Holder and Minkowski inequalities

Holder's inequality bounds the integral of a product by the product of Lp norms at conjugate exponents, and Minkowski's inequality establishes the triangle inequality for the Lp norm, the two estimates that make Lp a normed space.

Riesz-Fischer completeness theorem

Every Lp space is complete, so it is a Banach space and, for the exponent two, a Hilbert space; completeness is what links measure theory to functional analysis and underlies Fourier expansions.

Clinical relevance

Lp spaces are the natural setting for signals of finite energy and finite power, for the variational formulation of partial differential equations through Sobolev spaces, and for probability and statistics, where the space of square-integrable random variables carries the geometry behind variance, correlation, and least-squares estimation.

History

Riesz and Fischer independently proved the completeness of square-integrable functions in 1907, a result soon extended to general exponents. The Lp spaces became the prototype Banach spaces in Riesz's and Banach's development of functional analysis.

Key figures

• Frigyes Riesz
• Ernst Fischer
• Otto Holder

Seminal works

• folland1999
• brezis2011

Frequently asked questions

Why are Lp elements equivalence classes rather than functions?

The Lp norm cannot distinguish functions that differ only on a set of measure zero, so to obtain a genuine norm one identifies functions that agree almost everywhere and works with the resulting equivalence classes.

What is special about the case p equals two?

The space of square-integrable functions is a Hilbert space, the only Lp space with an inner product, which gives it orthogonality and projection and makes it the home of Fourier analysis and quantum states.