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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''.
 
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
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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 $1\le p < \infty$, 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
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\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 $\mu$.[[#References|[1]]]
 
where two functions $f$ and $g$ are equivalent if $f=g$ almost everywhere with respect to $\mu$.[[#References|[1]]]
  
The $L^p$ norm of $f$ for $1\le p \lt \infty$ is  
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The $L^p$ norm of $f$ for $1\le p < \infty$ is  
 
$$
 
$$
\| f \|_p  = \left( \int_S |f(s)|^p d\mu(s) < \infty \right)^{1/p}
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\| f \|_p  = \left( \int_S |f(s)|^p \; d\mu(s) < \infty \right)^{1/p}
 
$$
 
$$
  
For $p = \infty$,  $L^\infty$ space $L^p(S, F, \mu)$ consists of all the equivalence classes of measurable functions on $S$ such that for a positive constant $M < \infty$,
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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
 
|f(s)| < M
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almost everywhere with respect to $\mu$.[[#References|[1]]]
 
almost everywhere with respect to $\mu$.[[#References|[1]]]
  
For $0\lt p \lt 1$, the $L^p$ norm does not satisfy the triangle inequality.[[#References|[1]]]
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For $0 < p < 1$, the $L^p$ norm does not satisfy the triangle inequality.[[#References|[1]]]
  
 
== References ==
 
== References ==
[1] Stein, Elias M.; Shakarchi, Rami (2011). Functional Analysis: Introduction to Further Topics in Analysis. Chapter 1, Princeton University Press. ISBN 9780691113876.
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[1] Stein, Elias M.; Shakarchi, Rami (2011). Functional Analysis: Introduction to Further Topics in Analysis. Chapter 1, Princeton University Press. {{ISBN|9780691113876}}.

Latest revision as of 14:53, 11 November 2023

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 $1\le p < \infty$, 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 $1\le p < \infty$ 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 $0 < p < 1$, 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.

How to Cite This Entry:
Lp spaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lp_spaces&oldid=38877