Espacio LP

Function spaces generalizing finite-dimensional p norm spaces

En matemáticas , los espacios L ^p son espacios funcionales definidos mediante una generalización natural de la p -norma para espacios vectoriales de dimensión finita . A veces se denominan espacios de Lebesgue , en honor a Henri Lebesgue (Dunford & Schwartz 1958, III.3), aunque según el grupo de Bourbaki (Bourbaki 1987) fueron introducidos por primera vez por Frigyes Riesz (Riesz 1910).

Los espacios ^{de Lebesgue} forman una clase importante de espacios de Banach en el análisis funcional y de espacios vectoriales topológicos . Debido a su papel clave en el análisis matemático de espacios de medida y probabilidad, los espacios de Lebesgue también se utilizan en la discusión teórica de problemas en física, estadística, economía, finanzas, ingeniería y otras disciplinas.

Aplicaciones

Estadística

En estadística, las medidas de tendencia central y dispersión estadística , como la media , la mediana y la desviación estándar , pueden definirse en términos de métricas, y las medidas de tendencia central pueden caracterizarse como soluciones a problemas variacionales . ${\displaystyle L^{p}}$

En la regresión penalizada , la "penalización L1" y la "penalización L2" se refieren a penalizar la norma del vector de valores de parámetros de una solución (es decir, la suma de sus valores absolutos) o su norma al cuadrado (su longitud euclidiana ). Las técnicas que utilizan una penalización L1, como LASSO , fomentan las soluciones dispersas (donde los muchos parámetros son cero). ^[1] La regularización de red elástica utiliza un término de penalización que es una combinación de la norma y la norma al cuadrado del vector de parámetros. ${\displaystyle L^{1}}$ ${\displaystyle L^{2}}$ ${\displaystyle L^{1}}$ ${\displaystyle L^{2}}$

Desigualdad de Hausdorff-Young

La transformada de Fourier para la línea real (o, para funciones periódicas , véase serie de Fourier ), se asigna a (o a ) respectivamente, donde y Esto es una consecuencia del teorema de interpolación de Riesz-Thorin , y se precisa con la desigualdad de Hausdorff-Young . ${\displaystyle L^{p}(\mathbb {R} )}$ ${\displaystyle L^{q}(\mathbb {R} )}$ ${\displaystyle L^{p}(\mathbf {T} )}$ ${\displaystyle \ell ^{q}}$ ${\displaystyle 1\leq p\leq 2}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1.}$

Por el contrario, si la transformada de Fourier no se corresponde con ${\displaystyle p>2,}$ ${\displaystyle L^{q}.}$

Espacios de Hilbert

Los espacios de Hilbert son fundamentales para muchas aplicaciones, desde la mecánica cuántica hasta el cálculo estocástico . Los espacios y son ambos espacios de Hilbert. De hecho, al elegir una base de Hilbert , es decir, un subconjunto ortonormal máximo de o cualquier espacio de Hilbert, se ve que cada espacio de Hilbert es isométricamente isomorfo a (igual que arriba), es decir, un espacio de Hilbert de tipo ${\displaystyle L^{2}}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle E,}$ ${\displaystyle L^{2}}$ ${\displaystyle \ell ^{2}(E)}$ ${\displaystyle E}$ ${\displaystyle \ell ^{2}.}$

Elpag-norma en dimensiones finitas

Las ilustraciones de círculos unitarios (ver también superelipse ) se basan en diferentes normas (cada vector desde el origen hasta el círculo unitario tiene una longitud de uno, y la longitud se calcula con la fórmula de longitud del correspondiente ). ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle p}$ ${\displaystyle p}$

La longitud euclidiana de un vector en el espacio vectorial real -dimensional viene dada por la norma euclidiana : ${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ $${\displaystyle \|x\|_{2}=\left({x_{1}}^{2}+{x_{2}}^{2}+\dotsb +{x_{n}}^{2}\right)^{1/2}.}$$

La distancia euclidiana entre dos puntos y es la longitud de la línea recta entre los dos puntos. En muchas situaciones, la distancia euclidiana es apropiada para capturar las distancias reales en un espacio dado. Por el contrario, considere a los taxistas en un plano de calles en cuadrícula que deben medir la distancia no en términos de la longitud de la línea recta hasta su destino, sino en términos de la distancia rectilínea , que tiene en cuenta que las calles son ortogonales o paralelas entre sí. La clase de -normas generaliza estos dos ejemplos y tiene una gran cantidad de aplicaciones en muchas partes de las matemáticas , la física y la informática . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \|x-y\|_{2}}$ ${\displaystyle p}$

Definición

Para un número real, la -norma o -norma de se define por Las barras de valor absoluto se pueden eliminar cuando es un número racional con un numerador par en su forma reducida, y se extrae del conjunto de números reales, o uno de sus subconjuntos. ${\displaystyle p\geq 1,}$ ${\displaystyle p}$ ${\displaystyle L^{p}}$ ${\displaystyle x}$ $${\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.}$$ ${\displaystyle p}$ ${\displaystyle x}$

La norma euclidiana desde arriba cae en esta clase y es la -norma, y ​​la -norma es la norma que corresponde a la distancia rectilínea . ${\displaystyle 2}$ ${\displaystyle 1}$

La -norma o norma máxima (o norma uniforme) es el límite de las -normas para Resulta que este límite es equivalente a la siguiente definición: ${\displaystyle L^{\infty }}$ ${\displaystyle L^{p}}$ ${\displaystyle p\to \infty .}$ $${\displaystyle \|x\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|\right\}}$$

Ver L -infinito .

Porque todas las -normas y la norma máxima definidas anteriormente satisfacen de hecho las propiedades de una "función de longitud" (o norma ), que son: ${\displaystyle p\geq 1,}$ ${\displaystyle p}$

• sólo el vector cero tiene longitud cero,
• la longitud del vector es homogénea positiva con respecto a la multiplicación por un escalar ( homogeneidad positiva ), y
• la longitud de la suma de dos vectores no es mayor que la suma de las longitudes de los vectores ( desigualdad triangular ).

En términos abstractos, esto significa que junto con la -norma es un espacio vectorial normado . Además, resulta que este espacio es completo, por lo que es un espacio de Banach . Este espacio de Banach es el -espacio sobre ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle p}$ ${\displaystyle L^{p}}$ ${\displaystyle \{1,2,\ldots ,n\}.}$

Relaciones entrepag-normas

La distancia de cuadrícula o distancia rectilínea (a veces llamada " distancia de Manhattan ") entre dos puntos nunca es menor que la longitud del segmento de línea que los separa (distancia euclidiana o "distancia en línea recta"). Formalmente, esto significa que la norma euclidiana de cualquier vector está limitada por su norma 1: $${\displaystyle \|x\|_{2}\leq \|x\|_{1}.}$$

Este hecho se generaliza a las -normas en el sentido de que la -norma de cualquier vector dado no crece con : ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle \|x\|_{p}}$ ${\displaystyle x}$ ${\displaystyle p}$

${\displaystyle \|x\|_{p+a}\leq \|x\|_{p}}$para cualquier vector y números reales y (de hecho, esto sigue siendo cierto para y ). ${\displaystyle x}$ ${\displaystyle p\geq 1}$ ${\displaystyle a\geq 0.}$ ${\displaystyle 0<p<1}$ ${\displaystyle a\geq 0}$

Para la dirección opuesta, se conoce la siguiente relación entre la -norma y la -norma: ${\displaystyle 1}$ ${\displaystyle 2}$ $${\displaystyle \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}~.}$$

Esta desigualdad depende de la dimensión del espacio vectorial subyacente y se deriva directamente de la desigualdad de Cauchy-Schwarz . ${\displaystyle n}$

En general, para los vectores en donde ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle 0<r<p:}$ $${\displaystyle \|x\|_{p}\leq \|x\|_{r}\leq n^{{\frac {1}{r}}-{\frac {1}{p}}}\|x\|_{p}~.}$$

Esto es una consecuencia de la desigualdad de Hölder .

Cuando0 < p < 1

Astroide , círculo unitario en el sistema métrico ${\displaystyle p={\tfrac {2}{3}}}$

En la fórmula se define una función absolutamente homogénea para , sin embargo, la función resultante no define una norma, porque no es subaditiva . Por otra parte, la fórmula define una función subaditiva a costa de perder la homogeneidad absoluta. Sin embargo, sí define una F-norma , que es homogénea de grado ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n>1,}$ $${\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}\right)^{1/p}}$$ ${\displaystyle 0<p<1;}$ $${\displaystyle |x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}}$$ ${\displaystyle p.}$

Por lo tanto, la función define una métrica . El espacio métrico se denota por $${\displaystyle d_{p}(x,y)=\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}}$$ ${\displaystyle (\mathbb {R} ^{n},d_{p})}$ ${\displaystyle \ell _{n}^{p}.}$

Aunque la bola unitaria alrededor del origen en esta métrica es "cóncava", la topología definida por la métrica es la topología de espacio vectorial habitual de por lo tanto es un espacio vectorial topológico localmente convexo . Más allá de esta afirmación cualitativa, una forma cuantitativa de medir la falta de convexidad de es denotar por la constante más pequeña tal que el múltiplo escalar de la bola unitaria contenga la envoltura convexa de la cual es igual a El hecho de que para fijo tenemos muestra que el espacio de secuencia de dimensión infinita definido a continuación, ya no es localmente convexo. ^{[ cita requerida ]} ${\displaystyle p}$ ${\displaystyle B_{n}^{p}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle B_{p}}$ ${\displaystyle \mathbb {R} ^{n},}$ ${\displaystyle \ell _{n}^{p}}$ ${\displaystyle \ell _{n}^{p}}$ ${\displaystyle C_{p}(n)}$ ${\displaystyle C}$ ${\displaystyle C\,B_{n}^{p}}$ ${\displaystyle p}$ ${\displaystyle B_{n}^{p},}$ ${\displaystyle B_{n}^{1}.}$ ${\displaystyle p<1}$ $${\displaystyle C_{p}(n)=n^{{\tfrac {1}{p}}-1}\to \infty ,\quad {\text{as }}n\to \infty }$$ ${\displaystyle \ell ^{p}}$

Cuandop = 0

Hay una norma y otra función llamada "norma" (entre comillas). ${\displaystyle \ell _{0}}$ ${\displaystyle \ell _{0}}$

La definición matemática de la norma fue establecida por la Teoría de Operaciones Lineales de Banach . El espacio de sucesiones tiene una topología métrica completa proporcionada por la F-norma que es discutida por Stefan Rolewicz en Metric Linear Spaces . ^[2] El espacio -normado se estudia en análisis funcional, teoría de probabilidad y análisis armónico. ${\displaystyle \ell _{0}}$ $${\displaystyle (x_{n})\mapsto \sum _{n}2^{-n}{\frac {|x_{n}|}{1+|x_{n}|}},}$$ ${\displaystyle \ell _{0}}$

Otra función que David Donoho denominó "norma" (cuyas comillas advierten que esta función no es una norma propiamente dicha) es el número de entradas distintas de cero del vector ^{[ cita requerida ]} Muchos autores abusan de la terminología omitiendo las comillas. Definir la "norma" cero de es igual a ${\displaystyle \ell _{0}}$ ${\displaystyle x.}$ ${\displaystyle 0^{0}=0,}$ ${\displaystyle x}$ $${\displaystyle |x_{1}|^{0}+|x_{2}|^{0}+\cdots +|x_{n}|^{0}.}$$

Un gif animado de normas p de 0,1 a 2 con un paso de 0,05.

Esto no es una norma porque no es homogéneo . Por ejemplo, escalar el vector por una constante positiva no cambia la "norma". A pesar de estos defectos como norma matemática, la "norma" de conteo distinto de cero tiene usos en computación científica , teoría de la información y estadística , en particular en detección comprimida en procesamiento de señales y análisis armónico computacional . A pesar de no ser una norma, la métrica asociada, conocida como distancia de Hamming , es una distancia válida, ya que no se requiere homogeneidad para las distancias. ${\displaystyle x}$

Elpag-norma en dimensiones infinitas yℓ pespacios

El espacio de secuenciaℓ p

La -norma se puede extender a vectores que tienen un número infinito de componentes ( secuencias ), lo que produce el espacio Esto contiene como casos especiales: ${\displaystyle p}$ ${\displaystyle \ell ^{p}.}$

• ${\displaystyle \ell ^{1},}$el espacio de sucesiones cuyas series son absolutamente convergentes ,
• ${\displaystyle \ell ^{2},}$el espacio de secuencias cuadradamente sumables , que es un espacio de Hilbert , y
• ${\displaystyle \ell ^{\infty },}$el espacio de secuencias acotadas .

El espacio de sucesiones tiene una estructura natural de espacio vectorial al aplicar la suma y la multiplicación escalar coordenada por coordenada. Explícitamente, la suma vectorial y la acción escalar para sucesiones infinitas de números reales (o complejos ) vienen dadas por: $${\displaystyle {\begin{aligned}&(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots )+(y_{1},y_{2},\ldots ,y_{n},y_{n+1},\ldots )\\={}&(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n},x_{n+1}+y_{n+1},\ldots ),\\[6pt]&\lambda \cdot \left(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots \right)\\={}&(\lambda x_{1},\lambda x_{2},\ldots ,\lambda x_{n},\lambda x_{n+1},\ldots ).\end{aligned}}}$$

Defina la -norma: ${\displaystyle p}$ $${\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}+|x_{n+1}|^{p}+\cdots \right)^{1/p}}$$

Aquí surge una complicación, y es que la serie de la derecha no siempre es convergente, por lo que, por ejemplo, la sucesión formada sólo por unos, tendrá una -norma infinita para El espacio se define entonces como el conjunto de todas las sucesiones infinitas de números reales (o complejos) tales que la -norma es finita. ${\displaystyle (1,1,1,\ldots ),}$ ${\displaystyle p}$ ${\displaystyle 1\leq p<\infty .}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle p}$

Se puede comprobar que, a medida que aumenta, el conjunto se hace más grande. Por ejemplo, la sucesión no está en pero sí en para ya que la serie diverge para (la serie armónica ), pero es convergente para ${\displaystyle p}$ ${\displaystyle \ell ^{p}}$ $${\displaystyle \left(1,{\frac {1}{2}},\ldots ,{\frac {1}{n}},{\frac {1}{n+1}},\ldots \right)}$$ ${\displaystyle \ell ^{1},}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle p>1,}$ $${\displaystyle 1^{p}+{\frac {1}{2^{p}}}+\cdots +{\frac {1}{n^{p}}}+{\frac {1}{(n+1)^{p}}}+\cdots ,}$$ ${\displaystyle p=1}$ ${\displaystyle p>1.}$

También se define la -norma utilizando el supremo : y el espacio correspondiente de todas las sucesiones acotadas. Resulta que ^[3] si el lado derecho es finito, o el lado izquierdo es infinito. Por lo tanto, consideraremos espacios para ${\displaystyle \infty }$ $${\displaystyle \|x\|_{\infty }=\sup(|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|,|x_{n+1}|,\ldots )}$$ ${\displaystyle \ell ^{\infty }}$ $${\displaystyle \|x\|_{\infty }=\lim _{p\to \infty }\|x\|_{p}}$$ ${\displaystyle \ell ^{p}}$ ${\displaystyle 1\leq p\leq \infty .}$

La -norma así definida es, en efecto, una norma y, junto con esta norma, un espacio de Banach . El espacio totalmente general se obtiene, como se ve a continuación, considerando vectores, no sólo con un número finito o infinitamente numerable de componentes, sino con " un número arbitrario de componentes "; en otras palabras, funciones . Se utiliza una integral en lugar de una suma para definir la -norma. ${\displaystyle p}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle L^{p}}$ ${\displaystyle p}$

General ℓpag-espacio

En completa analogía con la definición precedente, se puede definir el espacio sobre un conjunto índice general (y ) como donde la convergencia a la derecha significa que solo un número contable de sumandos son distintos de cero (véase también Convergencia incondicional ). Con la norma, el espacio se convierte en un espacio de Banach. En el caso en que es finito con elementos, esta construcción da como resultado la -norma definida anteriormente. Si es contablemente infinito, este es exactamente el espacio de sucesiones definido anteriormente. Para conjuntos incontables , este es un espacio de Banach no separable que puede verse como el límite directo localmente convexo de los espacios de sucesiones. ^[4] ${\displaystyle \ell ^{p}(I)}$ ${\displaystyle I}$ ${\displaystyle 1\leq p<\infty }$ $${\displaystyle \ell ^{p}(I)=\left\{(x_{i})_{i\in I}\in \mathbb {K} ^{I}:\sum _{i\in I}|x_{i}|^{p}<+\infty \right\},}$$ $${\displaystyle \|x\|_{p}=\left(\sum _{i\in I}|x_{i}|^{p}\right)^{1/p}}$$ ${\displaystyle \ell ^{p}(I)}$ ${\displaystyle I}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle p}$ ${\displaystyle I}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle I}$ ${\displaystyle \ell ^{p}}$

Porque la -norma es incluso inducida por un producto interno canónico llamado ${\displaystyle p=2,}$ ${\displaystyle \|\,\cdot \,\|_{2}}$ ${\displaystyle \langle \,\cdot ,\,\cdot \rangle ,}$Producto interno euclidiano , lo que significa quese cumple para todos los vectores.Este producto interno se puede expresar en términos de la norma utilizando laidentidad de polarización. Ense puede definir por mientras que para el espacioasociado con unespacio de medidaque consta de todaslas funciones integrables al cuadrado, es ${\displaystyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}$ ${\displaystyle \mathbf {x} .}$ ${\displaystyle \ell ^{2},}$ $${\displaystyle \langle \left(x_{i}\right)_{i},\left(y_{n}\right)_{i}\rangle _{\ell ^{2}}~=~\sum _{i}x_{i}{\overline {y_{i}}}}$$ ${\displaystyle L^{2}(X,\mu )}$ ${\displaystyle (X,\Sigma ,\mu ),}$ $${\displaystyle \langle f,g\rangle _{L^{2}}=\int _{X}f(x){\overline {g(x)}}\,\mathrm {d} x.}$$

Consideremos ahora el caso Definir ^{[nota 1]} donde para todo ^{[5] [nota 2]} ${\displaystyle p=\infty .}$ $${\displaystyle \ell ^{\infty }(I)=\{x\in \mathbb {K} ^{I}:\sup \operatorname {range} |x|<+\infty \},}$$ ${\displaystyle x}$ $${\displaystyle \|x\|_{\infty }\equiv \inf\{C\in \mathbb {R} _{\geq 0}:|x_{i}|\leq C{\text{ for all }}i\in I\}={\begin{cases}\sup \operatorname {range} |x|&{\text{if }}X\neq \varnothing ,\\0&{\text{if }}X=\varnothing .\end{cases}}}$$

El conjunto de índices se puede convertir en un espacio de medida si se le asigna la σ-álgebra discreta y la medida de conteo . En ese caso, el espacio es simplemente un caso especial del espacio más general (definido a continuación). ${\displaystyle I}$ ${\displaystyle \ell ^{p}(I)}$ ${\displaystyle L^{p}}$

L pespacios e integrales de Lebesgue

Un espacio puede definirse como un espacio de funciones mensurables para las cuales la potencia -ésima del valor absoluto es integrable según Lebesgue , donde se identifican funciones que concuerdan casi en todas partes. De manera más general, sea un espacio de medida y ^{[nota 3]} Cuando , considere el conjunto de todas las funciones mensurables desde hasta o cuyo valor absoluto elevado a la potencia -ésima tiene una integral finita, o en símbolos: ${\displaystyle L^{p}}$ ${\displaystyle p}$ ${\displaystyle (S,\Sigma ,\mu )}$ ${\displaystyle 1\leq p\leq \infty .}$ ${\displaystyle p\neq \infty }$ ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ ${\displaystyle f}$ ${\displaystyle S}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle p}$ $${\displaystyle \|f\|_{p}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\int _{S}|f|^{p}\;\mathrm {d} \mu \right)^{1/p}<\infty .}$$

Para definir el conjunto recordemos que dos funciones y definidas en se dice que son iguales casi en todas partes , escrito ae , si el conjunto es medible y tiene medida cero. De manera similar, una función medible (y su valor absoluto ) está acotada (o dominada ) casi en todas partes por un número real escrito ae , si el conjunto (necesariamente) medible tiene medida cero. El espacio es el conjunto de todas las funciones mesurables que están acotadas casi en todas partes (por algún número real ) y se define como el ínfimo de estos límites: Cuando entonces este es el mismo que el supremo esencial del valor absoluto de : ^{[nota 4]} ${\displaystyle p=\infty ,}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle S}$ ${\displaystyle f=g}$ ${\displaystyle \{s\in S:f(s)\neq g(s)\}}$ ${\displaystyle f}$ ${\displaystyle C,}$ ${\displaystyle |f|\leq C}$ ${\displaystyle \{s\in S:|f(s)|>C\}}$ ${\displaystyle {\mathcal {L}}^{\infty }(S,\mu )}$ ${\displaystyle f}$ ${\displaystyle C}$ ${\displaystyle \|f\|_{\infty }}$ $${\displaystyle \|f\|_{\infty }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\inf\{C\in \mathbb {R} _{\geq 0}:|f(s)|\leq C{\text{ for almost every }}s\}.}$$ ${\displaystyle \mu (S)\neq 0}$ ${\displaystyle f}$ $${\displaystyle \|f\|_{\infty }~=~{\begin{cases}\operatorname {esssup} |f|&{\text{if }}\mu (S)>0,\\0&{\text{if }}\mu (S)=0.\end{cases}}}$$

Por ejemplo, si es una función medible que es igual a casi todas partes ^{[nota 5]} entonces para cada y por lo tanto para todos ${\displaystyle f}$ ${\displaystyle 0}$ ${\displaystyle \|f\|_{p}=0}$ ${\displaystyle p}$ ${\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu )}$ ${\displaystyle p.}$

Para cada positivo el valor bajo de una función medible y su valor absoluto son siempre los mismos (es decir, para todo ) y por lo tanto una función medible pertenece a si y solo si su valor absoluto lo es. Debido a esto, muchas fórmulas que involucran -normas se establecen solo para funciones de valor real no negativo. Consideremos por ejemplo la identidad que se cumple siempre que es medible, es real y (aquí cuando ). El requisito de no negatividad se puede eliminar sustituyendo en por lo que da Nótese en particular que cuando es finito entonces la fórmula relaciona la -norma con la -norma. ${\displaystyle p,}$ ${\displaystyle \|\,\cdot \,\|_{p}}$ ${\displaystyle f}$ ${\displaystyle |f|:S\to [0,\infty ]}$ ${\displaystyle \|f\|_{p}=\||f|\|_{p}}$ ${\displaystyle p}$ ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ ${\displaystyle p}$ ${\displaystyle \|f\|_{p}^{r}=\|f^{r}\|_{p/r},}$ ${\displaystyle f\geq 0}$ ${\displaystyle r>0}$ ${\displaystyle 0<p\leq \infty }$ ${\displaystyle \infty /r\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\infty }$ ${\displaystyle p=\infty }$ ${\displaystyle f\geq 0}$ ${\displaystyle |f|}$ ${\displaystyle f,}$ ${\displaystyle \|\,|f|\,\|_{p}^{r}=\|\,|f|^{r}\,\|_{p/r}.}$ ${\displaystyle p=r}$ ${\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}$ ${\displaystyle p}$ ${\displaystyle 1}$

Espacio semirregulado de funciones integrables de potencia -ésima ${\displaystyle p}$

Cada conjunto de funciones forma un espacio vectorial cuando la suma y la multiplicación escalar se definen puntualmente. ^{[nota 6]} Que la suma de dos funciones integrables en la potencia -ésima y es de nuevo integrable en la potencia -ésima se deduce de ^{[prueba 1]} aunque también es una consecuencia de la desigualdad de Minkowski que establece que satisface la desigualdad triangular para (la desigualdad triangular no se cumple para ). Que es cerrado bajo la multiplicación escalar se debe a que es absolutamente homogéneo , lo que significa que para cada escalar y cada función ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ ${\displaystyle p}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle p}$ ${\textstyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right),}$ $${\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}$$ ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle 1\leq p\leq \infty }$ ${\displaystyle 0<p<1}$ ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle \|sf\|_{p}=|s|\|f\|_{p}}$ ${\displaystyle s}$ ${\displaystyle f.}$

La homogeneidad absoluta , la desigualdad triangular y la no negatividad son las propiedades definitorias de una seminorma . Por lo tanto, es una seminorma y el conjunto de funciones integrables de potencia -ésima junto con la función define un espacio vectorial seminormado . En general, la seminorma no es una norma porque podrían existir funciones mensurables que satisfacen pero no son idénticamente iguales a ^{[nota 5]} ( es una norma si y solo si no existe tal ). ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ ${\displaystyle p}$ ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle f}$ ${\displaystyle \|f\|_{p}=0}$ ${\displaystyle 0}$ ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle f}$

Conjuntos cero de seminormas ${\displaystyle p}$

Si es medible e igual a ae entonces para todo positivo Por otro lado, si es una función medible para la cual existe alguna tal que entonces casi en todas partes. Cuando es finito entonces esto se sigue del caso y de la fórmula mencionada anteriormente. ${\displaystyle f}$ ${\displaystyle 0}$ ${\displaystyle \|f\|_{p}=0}$ ${\displaystyle p\leq \infty .}$ ${\displaystyle f}$ ${\displaystyle 0<p\leq \infty }$ ${\displaystyle \|f\|_{p}=0}$ ${\displaystyle f=0}$ ${\displaystyle p}$ ${\displaystyle p=1}$ ${\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}$

Por lo tanto, si es positivo y es cualquier función medible, entonces si y solo si casi en todas partes . Dado que el lado derecho ( ae ) no menciona , se deduce que todos tienen el mismo conjunto cero (no depende de ). Por lo tanto, denote este conjunto común por Este conjunto es un subespacio vectorial de para cada positivo ${\displaystyle p\leq \infty }$ ${\displaystyle f}$ ${\displaystyle \|f\|_{p}=0}$ ${\displaystyle f=0}$ ${\displaystyle f=0}$ ${\displaystyle p,}$ ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle p}$ $${\displaystyle {\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f:f=0\ \mu {\text{-almost everywhere}}\}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}\qquad \forall \ p.}$$ ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ ${\displaystyle p\leq \infty .}$

Espacio vectorial cociente

Al igual que cada seminorma , la seminorma induce una norma (definida en breve) en el espacio vectorial cociente canónico de por su subespacio vectorial. Este espacio vectorial cociente normado se denomina espacio de Lebesgue y es el tema de este artículo. Comenzamos definiendo el espacio vectorial cociente. ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ ${\textstyle {\mathcal {N}}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}.}$

Dado cualquier coconjunto , consta de todas las funciones medibles que son iguales a casi en todas partes . El conjunto de todos los coconjuntos, normalmente denotado por forma un espacio vectorial con origen cuando la suma vectorial y la multiplicación escalar se definen por y Este espacio vectorial cociente particular se denotará por ${\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}$ ${\displaystyle f+{\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f+h:h\in {\mathcal {N}}\}}$ ${\displaystyle g}$ ${\displaystyle f}$ $${\displaystyle {\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~~\{f+{\mathcal {N}}:f\in {\mathcal {L}}^{p}(S,\mu )\},}$$ ${\displaystyle 0+{\mathcal {N}}={\mathcal {N}}}$ ${\displaystyle (f+{\mathcal {N}})+(g+{\mathcal {N}})\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;(f+g)+{\mathcal {N}}}$ ${\displaystyle s(f+{\mathcal {N}})\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;(sf)+{\mathcal {N}}.}$ ${\displaystyle L^{p}(S,\,\mu )~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}.}$

Dos clases laterales son iguales si y solo si (o equivalentemente, ), lo que sucede si y solo si casi en todas partes; si este es el caso, entonces y se identifican en el espacio cociente. ${\displaystyle f+{\mathcal {N}}=g+{\mathcal {N}}}$ ${\displaystyle g\in f+{\mathcal {N}}}$ ${\displaystyle f-g\in {\mathcal {N}}}$ ${\displaystyle f=g}$ ${\displaystyle f}$ ${\displaystyle g}$

La -norma en el espacio vectorial cociente ${\displaystyle p}$

Dado cualquier valor de la seminorma en la clase lateral es constante e igual a denotar este valor único por de modo que: Esta asignación define un mapa, que también será denotado por en el espacio vectorial cociente. Este mapa es una norma en llamada ${\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}$ ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle f+{\mathcal {N}}=\{f+h:h\in {\mathcal {N}}\}}$ ${\displaystyle \|f\|_{p};}$ ${\displaystyle \|f+{\mathcal {N}}\|_{p},}$ $${\displaystyle \|f+{\mathcal {N}}\|_{p}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\|f\|_{p}.}$$ ${\displaystyle f+{\mathcal {N}}\mapsto \|f+{\mathcal {N}}\|_{p}}$ ${\displaystyle \|\cdot \|_{p},}$ $${\displaystyle L^{p}(S,\mu )~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~~{\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}~=~\{f+{\mathcal {N}}:f\in {\mathcal {L}}^{p}(S,\mu )\}.}$$ ${\displaystyle L^{p}(S,\mu )}$ ${\displaystyle p}$-norma . El valorde una clase laterales independiente de la función particularque se eligió para representarla, lo que significa que sies cualquier clase lateral entoncespara cada(ya quepara cada). ${\displaystyle \|f+{\mathcal {N}}\|_{p}}$ ${\displaystyle f+{\mathcal {N}}}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {C}}\in L^{p}(S,\mu )}$ ${\displaystyle \|{\mathcal {C}}\|_{p}=\|f\|_{p}}$ ${\displaystyle f\in {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}=f+{\mathcal {N}}}$ ${\displaystyle f\in {\mathcal {C}}}$

El espacio Lebesgue ${\displaystyle L^{p}}$

El espacio vectorial normado se denomina espacio o espacio de Lebesgue de funciones integrables de potencia -ésima y es un espacio de Banach para cada (lo que significa que es un espacio métrico completo , un resultado que a veces se denomina teorema de Riesz-Fischer ). Cuando se entiende el espacio de medida subyacente , a menudo se abrevia o incluso simplemente Dependiendo del autor, la notación de subíndice puede denotar uno o ${\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}$ ${\displaystyle L^{p}}$ ${\displaystyle p}$ ${\displaystyle 1\leq p\leq \infty }$ ${\displaystyle S}$ ${\displaystyle L^{p}(S,\mu )}$ ${\displaystyle L^{p}(\mu ),}$ ${\displaystyle L^{p}.}$ ${\displaystyle L_{p}}$ ${\displaystyle L^{p}(S,\mu )}$ ${\displaystyle L^{1/p}(S,\mu ).}$

Si la seminorma en resulta ser una norma (lo que sucede si y sólo si ) entonces el espacio normado será linealmente isométrico isomorfo al espacio cociente normado a través de la función canónica (ya que ); en otras palabras, serán, hasta una isometría lineal , el mismo espacio normado y por lo tanto ambos pueden ser llamados " espacio". ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ ${\displaystyle {\mathcal {N}}=\{0\}}$ ${\displaystyle \left({\mathcal {L}}^{p}(S,\,\mu ),\|\cdot \|_{p}\right)}$ ${\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}$ ${\displaystyle g\in {\mathcal {L}}^{p}(S,\,\mu )\mapsto \{g\}}$ ${\displaystyle g+{\mathcal {N}}=\{g\}}$ ${\displaystyle L^{p}}$

Las definiciones anteriores se generalizan a los espacios de Bochner .

En general, este proceso no se puede revertir: no hay una manera consistente de definir un representante "canónico" de cada clase lateral de en . Sin embargo, existe una teoría de ascensores que permite dicha recuperación. ${\displaystyle {\mathcal {N}}}$ ${\displaystyle L^{p}.}$ ${\displaystyle L^{\infty },}$

Casos especiales

Similar a los espacios, es el único espacio de Hilbert entre los espacios. En el caso complejo, el producto interno de se define por ${\displaystyle \ell ^{p}}$ ${\displaystyle L^{2}}$ ${\displaystyle L^{p}}$ ${\displaystyle L^{2}}$ $${\displaystyle \langle f,g\rangle =\int _{S}f(x){\overline {g(x)}}\,\mathrm {d} \mu (x)}$$

La estructura de producto interno adicional permite una teoría más rica, con aplicaciones, por ejemplo, a las series de Fourier y a la mecánica cuántica . Las funciones en a veces se denominan funciones integrables al cuadrado , funciones integrables cuadráticamente o funciones sumables al cuadrado , pero a veces estos términos se reservan para funciones que son integrables al cuadrado en algún otro sentido, como en el sentido de una integral de Riemann (Titchmarsh 1976). ${\displaystyle L^{2}}$

Si utilizamos funciones de valor complejo, el espacio es un álgebra C* conmutativa con multiplicación y conjugación puntuales. Para muchos espacios de medida, incluidos todos los sigma-finitos, es de hecho un álgebra de von Neumann conmutativa . Un elemento de define un operador acotado en cualquier espacio por multiplicación . ${\displaystyle L^{\infty }}$ ${\displaystyle L^{\infty }}$ ${\displaystyle L^{p}}$

Los espacios son un caso especial de espacios, cuando consisten en los números naturales y es la medida de conteo en De manera más general , si se considera cualquier conjunto con la medida de conteo, el espacio resultante se denota Por ejemplo, el espacio es el espacio de todas las secuencias indexadas por los números enteros, y al definir la -norma en dicho espacio, se suma sobre todos los números enteros. El espacio donde es el conjunto con elementos, es con su -norma como se definió anteriormente. Como cualquier espacio de Hilbert, cada espacio es linealmente isométrico a un adecuado donde la cardinalidad del conjunto es la cardinalidad de una base hilbertiana arbitraria para este particular ${\displaystyle 1\leq p\leq \infty }$ ${\displaystyle \ell ^{p}}$ ${\displaystyle L^{p}}$ ${\displaystyle S=\mathbb {N} )}$ ${\displaystyle \mu }$ ${\displaystyle \mathbb {N} .}$ ${\displaystyle S}$ ${\displaystyle L^{p}}$ ${\displaystyle \ell ^{p}(S).}$ ${\displaystyle \ell ^{p}(\mathbb {Z} )}$ ${\displaystyle p}$ ${\displaystyle \ell ^{p}(n),}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle p}$ ${\displaystyle L^{2}}$ ${\displaystyle \ell ^{2}(I),}$ ${\displaystyle I}$ ${\displaystyle L^{2}.}$

Propiedades deyopagespacios

Al igual que en el caso discreto, si existe tal que entonces ^{[ cita requerida ]} ${\displaystyle q<\infty }$ ${\displaystyle f\in L^{\infty }(S,\mu )\cap L^{q}(S,\mu ),}$ $${\displaystyle \|f\|_{\infty }=\lim _{p\to \infty }\|f\|_{p}.}$$

Desigualdad de Hölder

Supóngase que se satisface (donde ). Si y entonces y ^[6] ${\displaystyle p,q,r\in [1,\infty ]}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}={\tfrac {1}{r}}}$ ${\displaystyle {\tfrac {1}{\infty }}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~0}$ ${\displaystyle f\in L^{p}(S,\mu )}$ ${\displaystyle g\in L^{q}(S,\mu )}$ ${\displaystyle fg\in L^{r}(S,\mu )}$ $${\displaystyle \|fg\|_{r}~\leq ~\|f\|_{p}\,\|g\|_{q}.}$$

Esta desigualdad, llamada desigualdad de Hölder , es en cierto sentido óptima ^[6] ya que si (por lo que ) y es una función medible tal que donde el supremo se toma sobre la bola unitaria cerrada de entonces y ${\displaystyle r=1}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle f}$ $${\displaystyle \sup _{\|g\|_{q}\leq 1}\,\int _{S}|fg|\,\mathrm {d} \mu ~<~\infty }$$ ${\displaystyle L^{q}(S,\mu ),}$ ${\displaystyle f\in L^{p}(S,\mu )}$ $${\displaystyle \|f\|_{p}~=~\sup _{\|g\|_{q}\leq 1}\,\int _{S}fg\,\mathrm {d} \mu .}$$

Desigualdad de Minkowski

La desigualdad de Minkowski , que establece que satisface la desigualdad triangular , se puede generalizar: Si la función medible no es negativa (donde y son espacios de medida), entonces para todos los ^[7] ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle F:M\times N\to \mathbb {R} }$ ${\displaystyle (M,\mu )}$ ${\displaystyle (N,\nu )}$ ${\displaystyle 1\leq p\leq q\leq \infty ,}$ $${\displaystyle \left\|\left\|F(\,\cdot ,n)\right\|_{L^{p}(M,\mu )}\right\|_{L^{q}(N,\nu )}~\leq ~\left\|\left\|F(m,\cdot )\right\|_{L^{q}(N,\nu )}\right\|_{L^{p}(M,\mu )}\ .}$$

Descomposición atómica

If ${\displaystyle 1\leq p<\infty }$ then every non-negative ${\displaystyle f\in L^{p}(\mu )}$ has an atomic decomposition,^[8] meaning that there exist a sequence ${\displaystyle (r_{n})_{n\in \mathbb {Z} }}$ of non-negative real numbers and a sequence of non-negative functions ${\displaystyle (f_{n})_{n\in \mathbb {Z} },}$ called the atoms, whose supports ${\displaystyle \left(\operatorname {supp} f_{n}\right)_{n\in \mathbb {Z} }}$ are pairwise disjoint sets of measure ${\displaystyle \mu \left(\operatorname {supp} f_{n}\right)\leq 2^{n+1},}$ such that $${\displaystyle f~=~\sum _{n\in \mathbb {Z} }r_{n}\,f_{n}\,,}$$and for every integer ${\displaystyle n\in \mathbb {Z} ,}$ $${\displaystyle \|f_{n}\|_{\infty }~\leq ~2^{-{\tfrac {n}{p}}}\,,}$$ and $${\displaystyle {\tfrac {1}{2}}\|f\|_{p}^{p}~\leq ~\sum _{n\in \mathbb {Z} }r_{n}^{p}~\leq ~2\|f\|_{p}^{p}\,,}$$and where moreover, the sequence of functions ${\displaystyle (r_{n}f_{n})_{n\in \mathbb {Z} }}$ depends only on ${\displaystyle f}$ (it is independent of ${\displaystyle p}$).^[8] These inequalities guarantee that ${\displaystyle \|f_{n}\|_{p}^{p}\leq 2}$ for all integers ${\displaystyle n}$ while the supports of ${\displaystyle (f_{n})_{n\in \mathbb {Z} }}$ being pairwise disjoint implies^[8] $${\displaystyle \|f\|_{p}^{p}~=~\sum _{n\in \mathbb {Z} }r_{n}^{p}\,\|f_{n}\|_{p}^{p}\,.}$$

An atomic decomposition can be explicitly given by first defining for every integer ${\displaystyle n\in \mathbb {Z} ,}$^[8] $${\displaystyle t_{n}=\inf\{t\in \mathbb {R} :\mu (f>t)<2^{n}\}}$$(this infimum is attained by ${\displaystyle t_{n};}$ that is, ${\displaystyle \mu (f>t_{n})<2^{n}}$ holds) and then letting $${\displaystyle r_{n}~=~2^{n/p}\,t_{n}~{\text{ and }}\quad f_{n}~=~{\frac {f}{r_{n}}}\,\mathbf {1} _{(t_{n+1}<f\leq t_{n})}}$$where ${\displaystyle \mu (f>t)=\mu (\{s:f(s)>t\})}$ denotes the measure of the set ${\displaystyle (f>t):=\{s\in S:f(s)>t\}}$ and ${\displaystyle \mathbf {1} _{(t_{n+1}<f\leq t_{n})}}$ denotes the indicator function of the set ${\displaystyle (t_{n+1}<f\leq t_{n}):=\{s\in S:t_{n+1}<f(s)\leq t_{n}\}.}$ The sequence ${\displaystyle (t_{n})_{n\in \mathbb {Z} }}$ is decreasing and converges to ${\displaystyle 0}$ as ${\displaystyle n\to \infty .}$^[8] Consequently, if ${\displaystyle t_{n}=0}$ then ${\displaystyle t_{n+1}=0}$ and ${\displaystyle (t_{n+1}<f\leq t_{n})=\varnothing }$ so that ${\displaystyle f_{n}={\frac {1}{r_{n}}}\,f\,\mathbf {1} _{(t_{n+1}<f\leq t_{n})}}$ is identically equal to ${\displaystyle 0}$ (in particular, the division ${\displaystyle {\tfrac {1}{r_{n}}}}$ by ${\displaystyle r_{n}=0}$ causes no issues).

The complementary cumulative distribution function ${\displaystyle t\in \mathbb {R} \mapsto \mu (|f|>t)}$ of ${\displaystyle |f|=f}$ that was used to define the ${\displaystyle t_{n}}$ also appears in the definition of the weak ${\displaystyle L^{p}}$-norm (given below) and can be used to express the ${\displaystyle p}$-norm ${\displaystyle \|\cdot \|_{p}}$ (for ${\displaystyle 1\leq p<\infty }$) of ${\displaystyle f\in L^{p}(S,\mu )}$ as the integral^[8] $${\displaystyle \|f\|_{p}^{p}~=~p\,\int _{0}^{\infty }t^{p-1}\mu (|f|>t)\,\mathrm {d} t\,,}$$where the integration is with respect to the usual Lebesgue measure on ${\displaystyle (0,\infty ).}$

Dual spaces

The dual space (the Banach space of all continuous linear functionals) of ${\displaystyle L^{p}(\mu )}$ for ${\displaystyle 1<p<\infty }$ has a natural isomorphism with ${\displaystyle L^{q}(\mu ),}$ where ${\displaystyle q}$ is such that ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ (i.e. ${\displaystyle q={\tfrac {p}{p-1}}}$). This isomorphism associates ${\displaystyle g\in L^{q}(\mu )}$ with the functional ${\displaystyle \kappa _{p}(g)\in L^{p}(\mu )^{*}}$ defined by $${\displaystyle f\mapsto \kappa _{p}(g)(f)=\int fg\,\mathrm {d} \mu }$$ for every ${\displaystyle f\in L^{p}(\mu ).}$

The fact that ${\displaystyle \kappa _{p}(g)}$ is well defined and continuous follows from Hölder's inequality. ${\displaystyle \kappa _{p}:L^{q}(\mu )\to L^{p}(\mu )^{*}}$ is a linear mapping which is an isometry by the extremal case of Hölder's inequality. It is also possible to show (for example with the Radon–Nikodym theorem, see^[9]) that any ${\displaystyle G\in L^{p}(\mu )^{*}}$ can be expressed this way: i.e., that ${\displaystyle \kappa _{p}}$ is onto. Since ${\displaystyle \kappa _{p}}$ is onto and isometric, it is an isomorphism of Banach spaces. With this (isometric) isomorphism in mind, it is usual to say simply that ${\displaystyle L^{q}(\mu )}$ is the continuous dual space of ${\displaystyle L^{p}(\mu ).}$

For ${\displaystyle 1<p<\infty ,}$ the space ${\displaystyle L^{p}(\mu )}$ is reflexive. Let ${\displaystyle \kappa _{p}}$ be as above and let ${\displaystyle \kappa _{q}:L^{p}(\mu )\to L^{q}(\mu )^{*}}$ be the corresponding linear isometry. Consider the map from ${\displaystyle L^{p}(\mu )}$ to ${\displaystyle L^{p}(\mu )^{**},}$ obtained by composing ${\displaystyle \kappa _{q}}$ with the transpose (or adjoint) of the inverse of ${\displaystyle \kappa _{p}:}$

$${\displaystyle j_{p}:L^{p}(\mu )\mathrel {\overset {\kappa _{q}}{\longrightarrow }} L^{q}(\mu )^{*}\mathrel {\overset {\left(\kappa _{p}^{-1}\right)^{*}}{\longrightarrow }} L^{p}(\mu )^{**}}$$

This map coincides with the canonical embedding ${\displaystyle J}$ of ${\displaystyle L^{p}(\mu )}$ into its bidual. Moreover, the map ${\displaystyle j_{p}}$ is onto, as composition of two onto isometries, and this proves reflexivity.

If the measure ${\displaystyle \mu }$ on ${\displaystyle S}$ is sigma-finite, then the dual of ${\displaystyle L^{1}(\mu )}$ is isometrically isomorphic to ${\displaystyle L^{\infty }(\mu )}$ (more precisely, the map ${\displaystyle \kappa _{1}}$ corresponding to ${\displaystyle p=1}$ is an isometry from ${\displaystyle L^{\infty }(\mu )}$ onto ${\displaystyle L^{1}(\mu )^{*}.}$

The dual of ${\displaystyle L^{\infty }(\mu )}$ is subtler. Elements of ${\displaystyle L^{\infty }(\mu )^{*}}$ can be identified with bounded signed finitely additive measures on ${\displaystyle S}$ that are absolutely continuous with respect to ${\displaystyle \mu .}$ See ba space for more details. If we assume the axiom of choice, this space is much bigger than ${\displaystyle L^{1}(\mu )}$ except in some trivial cases. However, Saharon Shelah proved that there are relatively consistent extensions of Zermelo–Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of ${\displaystyle \ell ^{\infty }}$ is ${\displaystyle \ell ^{1}.}$^[10]

Embeddings

Colloquially, if ${\displaystyle 1\leq p<q\leq \infty ,}$ then ${\displaystyle L^{p}(S,\mu )}$ contains functions that are more locally singular, while elements of ${\displaystyle L^{q}(S,\mu )}$ can be more spread out. Consider the Lebesgue measure on the half line ${\displaystyle (0,\infty ).}$ A continuous function in ${\displaystyle L^{1}}$ might blow up near ${\displaystyle 0}$ but must decay sufficiently fast toward infinity. On the other hand, continuous functions in ${\displaystyle L^{\infty }}$ need not decay at all but no blow-up is allowed. The precise technical result is the following.^[11] Suppose that ${\displaystyle 0<p<q\leq \infty .}$ Then:

1. ${\displaystyle L^{q}(S,\mu )\subseteq L^{p}(S,\mu )}$ if and only if ${\displaystyle S}$ does not contain sets of finite but arbitrarily large measure (any finite measure, for example).
2. ${\displaystyle L^{p}(S,\mu )\subseteq L^{q}(S,\mu )}$ if and only if ${\displaystyle S}$ does not contain sets of non-zero but arbitrarily small measure (the counting measure, for example).

Neither condition holds for the real line with the Lebesgue measure while both conditions holds for the counting measure on any finite set. In both cases the embedding is continuous, in that the identity operator is a bounded linear map from ${\displaystyle L^{q}}$ to ${\displaystyle L^{p}}$ in the first case, and ${\displaystyle L^{p}}$ to ${\displaystyle L^{q}}$ in the second. (This is a consequence of the closed graph theorem and properties of ${\displaystyle L^{p}}$ spaces.) Indeed, if the domain ${\displaystyle S}$ has finite measure, one can make the following explicit calculation using Hölder's inequality $${\displaystyle \ \|\mathbf {1} f^{p}\|_{1}\leq \|\mathbf {1} \|_{q/(q-p)}\|f^{p}\|_{q/p}}$$leading to $${\displaystyle \ \|f\|_{p}\leq \mu (S)^{1/p-1/q}\|f\|_{q}.}$$

The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity ${\displaystyle I:L^{q}(S,\mu )\to L^{p}(S,\mu )}$ is precisely $${\displaystyle \|I\|_{q,p}=\mu (S)^{1/p-1/q}}$$the case of equality being achieved exactly when ${\displaystyle f=1}$ ${\displaystyle \mu }$-almost-everywhere.

Dense subspaces

Throughout this section we assume that ${\displaystyle 1\leq p<\infty .}$

Let ${\displaystyle (S,\Sigma ,\mu )}$ be a measure space. An integrable simple function ${\displaystyle f}$ on ${\displaystyle S}$ is one of the form $${\displaystyle f=\sum _{j=1}^{n}a_{j}\mathbf {1} _{A_{j}}}$$where ${\displaystyle a_{j}}$ are scalars, ${\displaystyle A_{j}\in \Sigma }$ has finite measure and ${\displaystyle {\mathbf {1} }_{A_{j}}}$ is the indicator function of the set ${\displaystyle A_{j},}$ for ${\displaystyle j=1,\dots ,n.}$ By construction of the integral, the vector space of integrable simple functions is dense in ${\displaystyle L^{p}(S,\Sigma ,\mu ).}$

More can be said when ${\displaystyle S}$ is a normal topological space and ${\displaystyle \Sigma }$ its Borel 𝜎–algebra, i.e., the smallest 𝜎–algebra of subsets of ${\displaystyle S}$ containing the open sets.

Suppose ${\displaystyle V\subseteq S}$ is an open set with ${\displaystyle \mu (V)<\infty .}$ It can be proved that for every Borel set ${\displaystyle A\in \Sigma }$ contained in ${\displaystyle V,}$ and for every ${\displaystyle \varepsilon >0,}$ there exist a closed set ${\displaystyle F}$ and an open set ${\displaystyle U}$ such that $${\displaystyle F\subseteq A\subseteq U\subseteq V\quad {\text{and}}\quad \mu (U)-\mu (F)=\mu (U\setminus F)<\varepsilon }$$

It follows that there exists a continuous Urysohn function ${\displaystyle 0\leq \varphi \leq 1}$ on ${\displaystyle S}$ that is ${\displaystyle 1}$ on ${\displaystyle F}$ and ${\displaystyle 0}$ on ${\displaystyle S\setminus U,}$ with $${\displaystyle \int _{S}|\mathbf {1} _{A}-\varphi |\,\mathrm {d} \mu <\varepsilon \,.}$$

If ${\displaystyle S}$ can be covered by an increasing sequence ${\displaystyle (V_{n})}$ of open sets that have finite measure, then the space of ${\displaystyle p}$–integrable continuous functions is dense in ${\displaystyle L^{p}(S,\Sigma ,\mu ).}$ More precisely, one can use bounded continuous functions that vanish outside one of the open sets ${\displaystyle V_{n}.}$

This applies in particular when ${\displaystyle S=\mathbb {R} ^{d}}$ and when ${\displaystyle \mu }$ is the Lebesgue measure. The space of continuous and compactly supported functions is dense in ${\displaystyle L^{p}(\mathbb {R} ^{d}).}$ Similarly, the space of integrable step functions is dense in ${\displaystyle L^{p}(\mathbb {R} ^{d});}$ this space is the linear span of indicator functions of bounded intervals when ${\displaystyle d=1,}$ of bounded rectangles when ${\displaystyle d=2}$ and more generally of products of bounded intervals.

Several properties of general functions in ${\displaystyle L^{p}(\mathbb {R} ^{d})}$ are first proved for continuous and compactly supported functions (sometimes for step functions), then extended by density to all functions. For example, it is proved this way that translations are continuous on ${\displaystyle L^{p}(\mathbb {R} ^{d}),}$ in the following sense: $${\displaystyle \forall f\in L^{p}\left(\mathbb {R} ^{d}\right):\quad \left\|\tau _{t}f-f\right\|_{p}\to 0,\quad {\text{as }}\mathbb {R} ^{d}\ni t\to 0,}$$where $${\displaystyle (\tau _{t}f)(x)=f(x-t).}$$

Closed subspaces

If ${\displaystyle 0<p<\infty }$ is any positive real number, ${\displaystyle \mu }$ is a probability measure on a measurable space ${\displaystyle (S,\Sigma )}$ (so that ${\displaystyle L^{\infty }(\mu )\subseteq L^{p}(\mu )}$), and ${\displaystyle V\subseteq L^{\infty }(\mu )}$ is a vector subspace, then ${\displaystyle V}$ is a closed subspace of ${\displaystyle L^{p}(\mu )}$ if and only if ${\displaystyle V}$ is finite-dimensional^[12] ( ${\displaystyle V}$ was chosen independent of ${\displaystyle p}$). In this theorem, which is due to Alexander Grothendieck,^[12] it is crucial that the vector space ${\displaystyle V}$ be a subset of ${\displaystyle L^{\infty }}$ since it is possible to construct an infinite-dimensional closed vector subspace of ${\displaystyle L^{1}\left(S^{1},{\tfrac {1}{2\pi }}\lambda \right)}$ (that is even a subset of ${\displaystyle L^{4}}$), where ${\displaystyle \lambda }$ is Lebesgue measure on the unit circle ${\displaystyle S^{1}}$ and ${\displaystyle {\tfrac {1}{2\pi }}\lambda }$ is the probability measure that results from dividing it by its mass ${\displaystyle \lambda (S^{1})=2\pi .}$^[12]

Lp (0 < p < 1)

Let ${\displaystyle (S,\Sigma ,\mu )}$ be a measure space. If ${\displaystyle 0<p<1,}$ then ${\displaystyle L^{p}(\mu )}$ can be defined as above: it is the quotient vector space of those measurable functions ${\displaystyle f}$ such that $${\displaystyle N_{p}(f)=\int _{S}|f|^{p}\,d\mu <\infty .}$$

As before, we may introduce the ${\displaystyle p}$-norm ${\displaystyle \|f\|_{p}=N_{p}(f)^{1/p},}$ but ${\displaystyle \|\cdot \|_{p}}$ does not satisfy the triangle inequality in this case, and defines only a quasi-norm. The inequality ${\displaystyle (a+b)^{p}\leq a^{p}+b^{p},}$ valid for ${\displaystyle a,b\geq 0,}$ implies that (Rudin 1991, §1.47) $${\displaystyle N_{p}(f+g)\leq N_{p}(f)+N_{p}(g)}$$and so the function $${\displaystyle d_{p}(f,g)=N_{p}(f-g)=\|f-g\|_{p}^{p}}$$is a metric on ${\displaystyle L^{p}(\mu ).}$ The resulting metric space is complete;^[13] the verification is similar to the familiar case when ${\displaystyle p\geq 1.}$The balls $${\displaystyle B_{r}=\{f\in L^{p}:N_{p}(f)<r\}}$$form a local base at the origin for this topology, as ${\displaystyle r>0}$ ranges over the positive reals.^[13] These balls satisfy ${\displaystyle B_{r}=r^{1/p}B_{1}}$ for all real ${\displaystyle r>0,}$ which in particular shows that ${\displaystyle B_{1}}$ is a bounded neighborhood of the origin;^[13] in other words, this space is locally bounded, just like every normed space, despite ${\displaystyle \|\cdot \|_{p}}$ not being a norm.

In this setting ${\displaystyle L^{p}}$ satisfies a reverse Minkowski inequality, that is for ${\displaystyle u,v\in L^{p}}$ $${\displaystyle {\Big \|}|u|+|v|{\Big \|}_{p}\geq \|u\|_{p}+\|v\|_{p}}$$

This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces ${\displaystyle L^{p}}$ for ${\displaystyle 1<p<\infty }$ (Adams & Fournier 2003).

The space ${\displaystyle L^{p}}$ for ${\displaystyle 0<p<1}$ is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: in ${\displaystyle \ell ^{p}}$ or ${\displaystyle L^{p}([0,1]),}$ every open convex set containing the ${\displaystyle 0}$ function is unbounded for the ${\displaystyle p}$-quasi-norm; therefore, the ${\displaystyle 0}$ vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space ${\displaystyle S}$ contains an infinite family of disjoint measurable sets of finite positive measure.

The only nonempty convex open set in ${\displaystyle L^{p}([0,1])}$ is the entire space (Rudin 1991, §1.47). As a particular consequence, there are no nonzero continuous linear functionals on ${\displaystyle L^{p}([0,1]);}$ the continuous dual space is the zero space. In the case of the counting measure on the natural numbers (producing the sequence space ${\displaystyle L^{p}(\mu )=\ell ^{p}}$), the bounded linear functionals on ${\displaystyle \ell ^{p}}$ are exactly those that are bounded on ${\displaystyle \ell ^{1},}$ namely those given by sequences in ${\displaystyle \ell ^{\infty }.}$ Although ${\displaystyle \ell ^{p}}$ does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.

The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. In the case of the Lebesgue measure on ${\displaystyle \mathbb {R} ^{n},}$ rather than work with ${\displaystyle L^{p}}$ for ${\displaystyle 0<p<1,}$ it is common to work with the Hardy space H^p whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the Hahn–Banach theorem still fails in H^p for ${\displaystyle p<1}$ (Duren 1970, §7.5).

L0, the space of measurable functions

The vector space of (equivalence classes of) measurable functions on ${\displaystyle (S,\Sigma ,\mu )}$ is denoted ${\displaystyle L^{0}(S,\Sigma ,\mu )}$ (Kalton, Peck & Roberts 1984). By definition, it contains all the ${\displaystyle L^{p},}$ and is equipped with the topology of convergence in measure. When ${\displaystyle \mu }$ is a probability measure (i.e., ${\displaystyle \mu (S)=1}$), this mode of convergence is named convergence in probability. The space ${\displaystyle L^{0}}$ is always a topological abelian group but is only a topological vector space if ${\displaystyle \mu (S)<\infty .}$ This is because scalar multiplication is continuous if and only if ${\displaystyle \mu (S)<\infty .}$ If ${\displaystyle (S,\Sigma ,\mu )}$ is ${\displaystyle \sigma }$-finite then the weaker topology of local convergence in measure is an F-space, i.e. a completely metrizable topological vector space. Moreover, this topology is isometric to global convergence in measure ${\displaystyle (S,\Sigma ,\nu )}$ for a suitable choice of probability measure ${\displaystyle \nu .}$

The description is easier when ${\displaystyle \mu }$ is finite. If ${\displaystyle \mu }$ is a finite measure on ${\displaystyle (S,\Sigma ),}$ the ${\displaystyle 0}$ function admits for the convergence in measure the following fundamental system of neighborhoods $${\displaystyle V_{\varepsilon }={\Bigl \{}f:\mu {\bigl (}\{x:|f(x)|>\varepsilon \}{\bigr )}<\varepsilon {\Bigr \}},\qquad \varepsilon >0.}$$

The topology can be defined by any metric ${\displaystyle d}$ of the form $${\displaystyle d(f,g)=\int _{S}\varphi {\bigl (}|f(x)-g(x)|{\bigr )}\,\mathrm {d} \mu (x)}$$where ${\displaystyle \varphi }$ is bounded continuous concave and non-decreasing on ${\displaystyle [0,\infty ),}$ with ${\displaystyle \varphi (0)=0}$ and ${\displaystyle \varphi (t)>0}$ when ${\displaystyle t>0}$ (for example, ${\displaystyle \varphi (t)=\min(t,1).}$ Such a metric is called Lévy-metric for ${\displaystyle L^{0}.}$ Under this metric the space ${\displaystyle L^{0}}$ is complete. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if ${\displaystyle \mu (S)<\infty }$. To see this, consider the Lebesgue measurable function ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$ defined by ${\displaystyle f(x)=x}$. Then clearly ${\displaystyle \lim _{c\rightarrow 0}d(cf,0)=\infty }$. The space ${\displaystyle L^{0}}$ is in general not locally bounded, and not locally convex.

For the infinite Lebesgue measure ${\displaystyle \lambda }$ on ${\displaystyle \mathbb {R} ^{n},}$ the definition of the fundamental system of neighborhoods could be modified as follows $${\displaystyle W_{\varepsilon }=\left\{f:\lambda \left(\left\{x:|f(x)|>\varepsilon {\text{ and }}|x|<{\tfrac {1}{\varepsilon }}\right\}\right)<\varepsilon \right\}}$$

The resulting space ${\displaystyle L^{0}(\mathbb {R} ^{n},\lambda )}$, with the topology of local convergence in measure, is isomorphic to the space ${\displaystyle L^{0}(\mathbb {R} ^{n},g\,\lambda ),}$ for any positive ${\displaystyle \lambda }$–integrable density ${\displaystyle g.}$

Generalizations and extensions

Weak Lp

Let ${\displaystyle (S,\Sigma ,\mu )}$ be a measure space, and ${\displaystyle f}$ a measurable function with real or complex values on ${\displaystyle S.}$ The distribution function of ${\displaystyle f}$ is defined for ${\displaystyle t\geq 0}$ by $${\displaystyle \lambda _{f}(t)=\mu \{x\in S:|f(x)|>t\}.}$$

If ${\displaystyle f}$ is in ${\displaystyle L^{p}(S,\mu )}$ for some ${\displaystyle p}$ with ${\displaystyle 1\leq p<\infty ,}$ then by Markov's inequality, $${\displaystyle \lambda _{f}(t)\leq {\frac {\|f\|_{p}^{p}}{t^{p}}}}$$

A function ${\displaystyle f}$ is said to be in the space weak ${\displaystyle L^{p}(S,\mu )}$, or ${\displaystyle L^{p,w}(S,\mu ),}$ if there is a constant ${\displaystyle C>0}$ such that, for all ${\displaystyle t>0,}$ $${\displaystyle \lambda _{f}(t)\leq {\frac {C^{p}}{t^{p}}}}$$

The best constant ${\displaystyle C}$ for this inequality is the ${\displaystyle L^{p,w}}$-norm of ${\displaystyle f,}$ and is denoted by $${\displaystyle \|f\|_{p,w}=\sup _{t>0}~t\lambda _{f}^{1/p}(t).}$$

The weak ${\displaystyle L^{p}}$ coincide with the Lorentz spaces ${\displaystyle L^{p,\infty },}$ so this notation is also used to denote them.

The ${\displaystyle L^{p,w}}$-norm is not a true norm, since the triangle inequality fails to hold. Nevertheless, for ${\displaystyle f}$ in ${\displaystyle L^{p}(S,\mu ),}$ $${\displaystyle \|f\|_{p,w}\leq \|f\|_{p}}$$and in particular ${\displaystyle L^{p}(S,\mu )\subset L^{p,w}(S,\mu ).}$

In fact, one has $${\displaystyle \|f\|_{L^{p}}^{p}=\int |f(x)|^{p}d\mu (x)\geq \int _{\{|f(x)|>t\}}t^{p}+\int _{\{|f(x)|\leq t\}}|f|^{p}\geq t^{p}\mu (\{|f|>t\}),}$$and raising to power ${\displaystyle 1/p}$ and taking the supremum in ${\displaystyle t}$ one has $${\displaystyle \|f\|_{L^{p}}\geq \sup _{t>0}t\;\mu (\{|f|>t\})^{1/p}=\|f\|_{L^{p,w}}.}$$

Under the convention that two functions are equal if they are equal ${\displaystyle \mu }$ almost everywhere, then the spaces ${\displaystyle L^{p,w}}$ are complete (Grafakos 2004).

For any ${\displaystyle 0<r<p}$ the expression $${\displaystyle \||f|\|_{L^{p,\infty }}=\sup _{0<\mu (E)<\infty }\mu (E)^{-1/r+1/p}\left(\int _{E}|f|^{r}\,d\mu \right)^{1/r}}$$is comparable to the ${\displaystyle L^{p,w}}$-norm. Further in the case ${\displaystyle p>1,}$ this expression defines a norm if ${\displaystyle r=1.}$ Hence for ${\displaystyle p>1}$ the weak ${\displaystyle L^{p}}$ spaces are Banach spaces (Grafakos 2004).

A major result that uses the ${\displaystyle L^{p,w}}$-spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.

Weighted Lp spaces

As before, consider a measure space ${\displaystyle (S,\Sigma ,\mu ).}$ Let ${\displaystyle w:S\to [a,\infty ),a>0}$ be a measurable function. The ${\displaystyle w}$-weighted ${\displaystyle L^{p}}$ space is defined as ${\displaystyle L^{p}(S,w\,\mathrm {d} \mu ),}$ where ${\displaystyle w\,\mathrm {d} \mu }$ means the measure ${\displaystyle \nu }$ defined by $${\displaystyle \nu (A)\equiv \int _{A}w(x)\,\mathrm {d} \mu (x),\qquad A\in \Sigma ,}$$

or, in terms of the Radon–Nikodym derivative, ${\displaystyle w={\tfrac {\mathrm {d} \nu }{\mathrm {d} \mu }}}$ the norm for ${\displaystyle L^{p}(S,w\,\mathrm {d} \mu )}$ is explicitly $${\displaystyle \|u\|_{L^{p}(S,w\,\mathrm {d} \mu )}\equiv \left(\int _{S}w(x)|u(x)|^{p}\,\mathrm {d} \mu (x)\right)^{1/p}}$$

As ${\displaystyle L^{p}}$-spaces, the weighted spaces have nothing special, since ${\displaystyle L^{p}(S,w\,\mathrm {d} \mu )}$ is equal to ${\displaystyle L^{p}(S,\mathrm {d} \nu ).}$ But they are the natural framework for several results in harmonic analysis (Grafakos 2004); they appear for example in the Muckenhoupt theorem: for ${\displaystyle 1<p<\infty ,}$ the classical Hilbert transform is defined on ${\displaystyle L^{p}(\mathbf {T} ,\lambda )}$ where ${\displaystyle \mathbf {T} }$ denotes the unit circle and ${\displaystyle \lambda }$ the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on ${\displaystyle L^{p}(\mathbb {R} ^{n},\lambda ).}$ Muckenhoupt's theorem describes weights ${\displaystyle w}$ such that the Hilbert transform remains bounded on ${\displaystyle L^{p}(\mathbf {T} ,w\,\mathrm {d} \lambda )}$ and the maximal operator on ${\displaystyle L^{p}(\mathbb {R} ^{n},w\,\mathrm {d} \lambda ).}$

Lp spaces on manifolds

One may also define spaces ${\displaystyle L^{p}(M)}$ on a manifold, called the intrinsic ${\displaystyle L^{p}}$ spaces of the manifold, using densities.

Vector-valued Lp spaces

Given a measure space ${\displaystyle (\Omega ,\Sigma ,\mu )}$ and a locally convex space ${\displaystyle E}$ (here assumed to be complete), it is possible to define spaces of ${\displaystyle p}$-integrable ${\displaystyle E}$-valued functions on ${\displaystyle \Omega }$ in a number of ways. One way is to define the spaces of Bochner integrable and Pettis integrable functions, and then endow them with locally convex TVS-topologies that are (each in their own way) a natural generalization of the usual ${\displaystyle L^{p}}$ topology. Another way involves topological tensor products of ${\displaystyle L^{p}(\Omega ,\Sigma ,\mu )}$ with ${\displaystyle E.}$ Element of the vector space ${\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes E}$ are finite sums of simple tensors ${\displaystyle f_{1}\otimes e_{1}+\cdots +f_{n}\otimes e_{n},}$ where each simple tensor ${\displaystyle f\times e}$ may be identified with the function ${\displaystyle \Omega \to E}$ that sends ${\displaystyle x\mapsto ef(x).}$ This tensor product ${\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes E}$ is then endowed with a locally convex topology that turns it into a topological tensor product, the most common of which are the projective tensor product, denoted by ${\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes _{\pi }E,}$ and the injective tensor product, denoted by ${\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes _{\varepsilon }E.}$ In general, neither of these space are complete so their completions are constructed, which are respectively denoted by ${\displaystyle L^{p}(\Omega ,\Sigma ,\mu ){\widehat {\otimes }}_{\pi }E}$ and ${\displaystyle L^{p}(\Omega ,\Sigma ,\mu ){\widehat {\otimes }}_{\varepsilon }E}$ (this is analogous to how the space of scalar-valued simple functions on ${\displaystyle \Omega ,}$ when seminormed by any ${\displaystyle \|\cdot \|_{p},}$ is not complete so a completion is constructed which, after being quotiented by ${\displaystyle \ker \|\cdot \|_{p},}$ is isometrically isomorphic to the Banach space ${\displaystyle L^{p}(\Omega ,\mu )}$). Alexander Grothendieck showed that when ${\displaystyle E}$ is a nuclear space (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable.

See also

• Bochner space – Type of topological space
• Orlicz space – Type of function space
• Hardy space – Concept within complex analysis
• Riesz–Thorin theorem – Theorem on operator interpolation
• Hölder mean – N-th root of the arithmetic mean of the given numbers raised to the power nPages displaying short descriptions of redirect targets
• Hölder space – Type of continuity of a complex-valued functionPages displaying short descriptions of redirect targets
• Root mean square – Square root of the mean square
• Least absolute deviations – Statistical optimality criterion
• Locally integrable function – Function which is integratable on its domain ${\displaystyle \left(L_{\text{loc}}^{1}\right)}$
• ${\displaystyle L^{p}(G)}$ spaces over a locally compact group ${\displaystyle G}$ – Duality for locally compact abelian groups
• Least-squares spectral analysis – Periodicity computation method
• List of Banach spaces
• Minkowski distance – Mathematical metric in normed vector space
• L-infinity – Space of bounded sequences
• L^p sum – Measure in functional analysis

Notes

1. Hastie, T. J.; Tibshirani, R.; Wainwright, M. J. (2015). Statistical Learning with Sparsity: The Lasso and Generalizations. CRC Press. ISBN 978-1-4987-1216-3.
2. Rolewicz, Stefan (1987), Functional analysis and control theory: Linear systems, Mathematics and its Applications (East European Series), vol. 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers, pp. xvi+524, doi:10.1007/978-94-015-7758-8, ISBN 90-277-2186-6, MR 0920371, OCLC 13064804^{[page needed]}
3. Maddox, I. J. (1988), Elements of Functional Analysis (2nd ed.), Cambridge: CUP, page 16
4. Rafael Dahmen, Gábor Lukács: Long colimits of topological groups I: Continuous maps and homeomorphisms. in: Topology and its Applications Nr. 270, 2020. Example 2.14
5. Garling, D. J. H. (2007). Inequalities: A Journey into Linear Analysis. Cambridge University Press. p. 54. ISBN 978-0-521-87624-7.
6. Bahouri, Chemin & Danchin 2011, pp. 1–4.
7. Bahouri, Chemin & Danchin 2011, p. 4.
8. Bahouri, Chemin & Danchin 2011, pp. 7–8.
9. Rudin, Walter (1980), Real and Complex Analysis (2nd ed.), New Delhi: Tata McGraw-Hill, ISBN 9780070542341, Theorem 6.16
10. Schechter, Eric (1997), Handbook of Analysis and its Foundations, London: Academic Press Inc. See Sections 14.77 and 27.44–47
11. Villani, Alfonso (1985), "Another note on the inclusion L^p(μ) ⊂ L^q(μ)", Amer. Math. Monthly, 92 (7): 485–487, doi:10.2307/2322503, JSTOR 2322503, MR 0801221
12. Rudin 1991, pp. 117–119.
13. Rudin 1991, p. 37.

1. The condition ${\displaystyle \sup \operatorname {range} |x|<+\infty .}$ is not equivalent to ${\displaystyle \sup \operatorname {range} |x|}$ being finite, unless ${\displaystyle X\neq \varnothing .}$
2. If ${\displaystyle X=\varnothing }$ then ${\displaystyle \sup \operatorname {range} |x|=-\infty .}$
3. The definitions of ${\displaystyle \|\cdot \|_{p},}$ ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu ),}$ and ${\displaystyle L^{p}(S,\,\mu )}$ can be extended to all ${\displaystyle 0<p\leq \infty }$ (rather than just ${\displaystyle 1\leq p\leq \infty }$), but it is only when ${\displaystyle 1\leq p\leq \infty }$ that ${\displaystyle \|\cdot \|_{p}}$ is guaranteed to be a norm (although ${\displaystyle \|\cdot \|_{p}}$ is a quasi-seminorm for all ${\displaystyle 0<p\leq \infty ,}$).
4. If ${\displaystyle \mu (S)=0}$ then ${\displaystyle \operatorname {esssup} |f|=-\infty .}$
5. For example, if a non-empty measurable set ${\displaystyle N\neq \varnothing }$ of measure ${\displaystyle \mu (N)=0}$ exists then its indicator function ${\displaystyle \mathbf {1} _{N}}$ satisfies ${\displaystyle \|\mathbf {1} _{N}\|_{p}=0}$ although ${\displaystyle \mathbf {1} _{N}\neq 0.}$
6. Explicitly, the vector space operations are defined by: $${\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x),\\(sf)(x)&=sf(x)\end{aligned}}}$$for all ${\displaystyle f,g\in {\mathcal {L}}^{p}(S,\,\mu )}$ and all scalars ${\displaystyle s.}$ These operations make ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ into a vector space because if ${\displaystyle s}$ is any scalar and ${\displaystyle f,g\in {\mathcal {L}}^{p}(S,\,\mu )}$ then both ${\displaystyle sf}$ and ${\displaystyle f+g}$ also belong to ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu ).}$

1. When ${\displaystyle 1\leq p<\infty ,}$ the inequality ${\displaystyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right)}$ can be deduced from the fact that the function ${\displaystyle F:[0,\infty )\to \mathbb {R} }$ defined by ${\displaystyle F(t)=t^{p}}$ is convex, which by definition means that ${\displaystyle F(tx+(1-t)y)\leq tF(x)+(1-t)F(y)}$ for all ${\displaystyle 0\leq t\leq 1}$ and all ${\displaystyle x,y}$ in the domain of ${\displaystyle F.}$ Substituting ${\displaystyle |f|,|g|,}$ and ${\displaystyle {\tfrac {1}{2}}}$ in for ${\displaystyle x,y,}$ and ${\displaystyle t}$ gives ${\displaystyle \left({\tfrac {1}{2}}|f|+{\tfrac {1}{2}}|g|\right)^{p}\leq {\tfrac {1}{2}}|f|^{p}+{\tfrac {1}{2}}|g|^{p},}$ which proves that ${\displaystyle (|f|+|g|)^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}$ The triangle inequality ${\displaystyle |f+g|\leq |f|+|g|}$ now implies ${\displaystyle |f+g|^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}$ The desired inequality follows by integrating both sides. ${\displaystyle \blacksquare }$

References

• Adams, Robert A.; Fournier, John F. (2003), Sobolev Spaces (Second ed.), Academic Press, ISBN 978-0-12-044143-3.
• Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. ISBN 978-3-642-16830-7. OCLC 704397128.
• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-13627-9.
• DiBenedetto, Emmanuele (2002), Real analysis, Birkhäuser, ISBN 3-7643-4231-5.
• Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume I, Wiley-Interscience.
• Duren, P. (1970), Theory of H^p-Spaces, New York: Academic Press
• Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Pearson Education, Inc., pp. 253–257, ISBN 0-13-035399-X.
• Hewitt, Edwin; Stromberg, Karl (1965), Real and abstract analysis, Springer-Verlag.
• Kalton, Nigel J.; Peck, N. Tenney; Roberts, James W. (1984), An F-space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511662447, ISBN 0-521-27585-7, MR 0808777
• Riesz, Frigyes (1910), "Untersuchungen über Systeme integrierbarer Funktionen", Mathematische Annalen, 69 (4): 449–497, doi:10.1007/BF01457637, S2CID 120242933
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157
• Titchmarsh, EC (1976), The theory of functions, Oxford University Press, ISBN 978-0-19-853349-8

External links

• "Lebesgue space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Proof that Lp spaces are complete