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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b1200401.png" /> be a complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b1200402.png" />-finite [[Measure space|measure space]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b1200403.png" /> be the space of all equivalence classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b1200404.png" />-measurable real-valued functions endowed with the topology of convergence in measure relative to each set of finite measure.
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A [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b1200405.png" /> is called a Banach function space on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b1200406.png" /> if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b1200407.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b1200408.png" /> almost everywhere and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b1200409.png" /> satisfies the ideal property:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004010.png" /></td> </tr></table>
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Let $( \Omega , \Sigma , \mu )$ be a complete $\sigma$-finite [[Measure space|measure space]] and let $L ^ { 0 } ( \mu ) = L ^ { 0 } ( \Omega , \Sigma , \mu )$ be the space of all equivalence classes of $\mu$-measurable real-valued functions endowed with the topology of convergence in measure relative to each set of finite measure.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004011.png" /></td> </tr></table>
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A [[Banach space|Banach space]] $X \subset L ^ { 0 } ( \mu )$ is called a Banach function space on $( \Omega , \Sigma , \mu )$ if there exists a $u \in X$ such that $u &gt; 0$ almost everywhere and $X$ satisfies the ideal property:
  
The Lebesgue function spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004013.png" />) play a primary role in many problems arising in mathematical analysis. There are other classes of Banach function spaces that are also of interest. The classes of Musielak–Orlicz, Lorentz and Marcinkiewicz spaces, for example, are of intrinsic importance (cf. also [[Orlicz space|Orlicz space]]; [[Orlicz–Lorentz space|Orlicz–Lorentz space]]; [[Marcinkiewicz space|Marcinkiewicz space]]). Function spaces are important and natural examples of abstract Banach lattices (a Banach lattice is a Banach space that is also a [[Vector lattice|vector lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004014.png" /> with the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004015.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004017.png" />, cf. also [[Banach lattice|Banach lattice]]). A Banach lattice is said to be order continuous if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004018.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004019.png" />. The following very useful general representation result (see [[#References|[a12]]]) allows one to reduce most of the proofs for a quite large class of abstract Banach lattices to the case of Banach function spaces: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004020.png" /> be an order-continuous Banach lattice with a weak unit (a weak unit is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004022.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004023.png" />). Then there exist a [[Probability space|probability space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004024.png" /> and a Banach function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004027.png" /> is isometrically lattice-isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004029.png" /> with continuous inclusions.
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\begin{equation*} x \in L ^ { 0 } ( \mu ) , y \in X , | x | \leq | y | \mu - a.e. \end{equation*}
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\begin{equation*} \Downarrow x \in X \text { and } \| x \| \leq \| y \|. \end{equation*}
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The Lebesgue function spaces $L _ { p }$ ($1 \leq p \leq \infty$) play a primary role in many problems arising in mathematical analysis. There are other classes of Banach function spaces that are also of interest. The classes of Musielak–Orlicz, Lorentz and Marcinkiewicz spaces, for example, are of intrinsic importance (cf. also [[Orlicz space|Orlicz space]]; [[Orlicz–Lorentz space|Orlicz–Lorentz space]]; [[Marcinkiewicz space|Marcinkiewicz space]]). Function spaces are important and natural examples of abstract Banach lattices (a Banach lattice is a Banach space that is also a [[Vector lattice|vector lattice]] $X$ with the property that $\| x \| \leq \| y \|$ whenever $| x | \leq | y |$, where $| x | = x \vee ( - x )$, cf. also [[Banach lattice|Banach lattice]]). A Banach lattice is said to be order continuous if $\| x _ { n } \| \rightarrow 0$ whenever $x _ { n } \downarrow 0$. The following very useful general representation result (see [[#References|[a12]]]) allows one to reduce most of the proofs for a quite large class of abstract Banach lattices to the case of Banach function spaces: Let $X$ be an order-continuous Banach lattice with a weak unit (a weak unit is an element $e &gt; 0$ such that $e \wedge | x | = 0$ implies $x = 0$). Then there exist a [[Probability space|probability space]] $( \Omega , \Sigma , \mu )$ and a Banach function space $X$ on $( \Omega , \Sigma , \mu )$ such that $X$ is isometrically lattice-isomorphic to $X$ and $L _ { \infty } ( \mu ) \subset X \subset L _ { 1 } ( \mu )$ with continuous inclusions.
  
 
See [[#References|[a2]]], [[#References|[a7]]], [[#References|[a10]]], [[#References|[a14]]] for a general theory of Banach lattices.
 
See [[#References|[a2]]], [[#References|[a7]]], [[#References|[a10]]], [[#References|[a14]]] for a general theory of Banach lattices.
  
A Banach function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004030.png" /> is said to have the Fatou property if whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004031.png" /> is a norm-bounded sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004033.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004035.png" />.
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A Banach function space $X$ is said to have the Fatou property if whenever $( f _ { n } )$ is a norm-bounded sequence in $X$ such that $0 \leq f _ { n } \uparrow f \in L ^ { 0 } ( \mu )$, then $f \in X$ and $\| f _ { n } \| \rightarrow \| f \|$.
  
In recent (1998) years a great deal of research went into the study of rearrangement-invariant function spaces, in particular of Orlicz spaces. General references to this area are e.g. [[#References|[a7]]], [[#References|[a11]]], [[#References|[a12]]]. A Banach function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004036.png" /> is said to be rearrangement invariant if whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004038.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004040.png" /> are equi-measurable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004042.png" />. Two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004044.png" /> are called equi-measurable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004046.png" /> have identical distributions, that is,
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In recent (1998) years a great deal of research went into the study of rearrangement-invariant function spaces, in particular of Orlicz spaces. General references to this area are e.g. [[#References|[a7]]], [[#References|[a11]]], [[#References|[a12]]]. A Banach function space $X$ is said to be rearrangement invariant if whenever $f \in X$, $g \in L ^ { 0 } ( \mu )$, and $f$ and $g$ are equi-measurable, then $g \in X$ and $\| f \| = \| g \|$. Two functions $f$ and $g$ are called equi-measurable if $| f|$ and $| g |$ have identical distributions, that is,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004047.png" /></td> </tr></table>
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\begin{equation*} \mu _ { f } ( \lambda ) = \mu \{ t \in \Omega : | f ( t ) | &gt; \lambda \} = \mu _ { g } ( \lambda ) \end{equation*}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004048.png" />.
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for all $\lambda &gt; 0$.
  
In the study of rearrangement-invariant function spaces, the Boyd indices play an important role (see e.g. [[#References|[a7]]], [[#References|[a12]]], and [[Boyd index|Boyd index]]). The Boyd indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004050.png" /> of a rearrangement-invariant function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004051.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004052.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004053.png" /> are defined by
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In the study of rearrangement-invariant function spaces, the Boyd indices play an important role (see e.g. [[#References|[a7]]], [[#References|[a12]]], and [[Boyd index|Boyd index]]). The Boyd indices $p_{X} $ and $q_X$ of a rearrangement-invariant function space $X$ on $[0,1]$ or $[ 0 , \infty )$ are defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004054.png" /></td> </tr></table>
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\begin{equation*} p _ { X } = \operatorname { lim } _ { s \rightarrow \infty } \frac { \operatorname { log } s } { \operatorname { log } \| D _ { s } \| _ { X } }, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004055.png" /></td> </tr></table>
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\begin{equation*} q _ { X } = \operatorname { lim } _ { s \rightarrow 0 + } \frac { \operatorname { log } s } { \operatorname { log } \| D _ { s } \| _ { X } }, \end{equation*}
  
where for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004057.png" /> denotes the dilation operator, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004058.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004059.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004060.png" /> is defined to be zero outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004061.png" /> in the former case).
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where for $s &gt; 0$, $D _ { S }$ denotes the dilation operator, defined by $D _ { s } f ( t ) = f ( t / s )$ for $f \in X$ (where $f$ is defined to be zero outside $[0,1]$ in the former case).
  
For example, consider the following results, which hold for every separable rearrangement-invariant function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004062.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004063.png" />:
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For example, consider the following results, which hold for every separable rearrangement-invariant function space $X$ on $[0,1]$:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004064.png" /> has an unconditional basis if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004066.png" /> (see, e.g., [[#References|[a11]]], [[#References|[a12]]]);
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i) $X$ has an unconditional basis if and only if $1 &lt; p_{ X}$ and $q_{X} &lt; \infty$ (see, e.g., [[#References|[a11]]], [[#References|[a12]]]);
  
ii) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004068.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004069.png" /> is a primary, i.e., whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004070.png" />, then at least one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004072.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004073.png" /> (see [[#References|[a3]]]).
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ii) if $1 &lt; p_{ X}$ and $q_{X} &lt; \infty$, then $X$ is a primary, i.e., whenever $X = E \oplus F$, then at least one of $E$ and $F$ is isomorphic to $X$ (see [[#References|[a3]]]).
  
Rearrangement-invariant function spaces play an important role in the theory of [[Interpolation of operators|interpolation of operators]] (see [[#References|[a4]]], [[#References|[a11]]]). A remarkable result of A.P. Calderón [[#References|[a5]]] on the characterization of all interpolation spaces between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004075.png" /> asserts that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004076.png" /> is an interpolation space with respect to the couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004077.png" /> (i.e., that every [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004078.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004080.png" /> boundedly, also maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004081.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004082.png" /> boundedly) if and only if it has the following property: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004083.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004084.png" />, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004085.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004086.png" />, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004088.png" /> for some absolute constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004089.png" />.
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Rearrangement-invariant function spaces play an important role in the theory of [[Interpolation of operators|interpolation of operators]] (see [[#References|[a4]]], [[#References|[a11]]]). A remarkable result of A.P. Calderón [[#References|[a5]]] on the characterization of all interpolation spaces between $L _ { 1 } = L _ { 1 } ( \mu )$ and $L _ { \infty } = L _ { \infty } ( \mu )$ asserts that $X$ is an interpolation space with respect to the couple $( L _ { 1 } , L _ { \infty } )$ (i.e., that every [[Linear operator|linear operator]] $T : L _ { 1 } + L _ { \infty } \rightarrow L _ { 1 } + L _ { \infty }$ such that $T : L _ { 1 } \rightarrow L _ { 1 }$ and $T : L _ { \infty } \rightarrow L _ { \infty }$ boundedly, also maps $X$ to $X$ boundedly) if and only if it has the following property: For every $g \in X$ and every $f \in L _ { 1 } + L _ { \infty }$, whenever $\int _ { 0 } ^ { t } f ^ { * } ( s ) d s \leq \int _ { 0 } ^ { t } g ^ { * } ( s ) d s$ for all $t &gt; 0$, it follows that $f \in X$ and $\| f \|_X \leq C\| g  \|_X$ for some absolute constant $C$.
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004090.png" /> denotes the non-increasing rearrangement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004091.png" />, which is defined by
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Here, $f ^ { * }$ denotes the non-increasing rearrangement of $f$, which is defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004092.png" /></td> </tr></table>
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\begin{equation*} f ^ { * } ( t ) = \operatorname { inf } \{ \lambda &gt; 0 : \mu _ { f } ( \lambda ) \leq t \} \end{equation*}
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004093.png" />. In particular, Calderón's result implies that rearrangement-invariant function spaces which have the Fatou property or are separable are interpolation spaces between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004095.png" />.
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for $t &gt; 0$. In particular, Calderón's result implies that rearrangement-invariant function spaces which have the Fatou property or are separable are interpolation spaces between $L_1$ and $L _ { \infty }$.
  
The Köthe dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004096.png" /> of a Banach function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004097.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004098.png" /> is defined to be the space of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b12004099.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040100.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040101.png" /> (cf. also [[Köthe–Toeplitz dual|Köthe–Toeplitz dual]]). The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040102.png" /> is a Banach function space endowed with the norm
+
The Köthe dual space $X ^ { \prime }$ of a Banach function space $X$ on $( \Omega , \Sigma , \mu )$ is defined to be the space of all $x ^ { \prime }$ for which $x x ^ { \prime } \in L _ { 1 } ( \mu )$ for each $x \in X$ (cf. also [[Köthe–Toeplitz dual|Köthe–Toeplitz dual]]). The space $X ^ { \prime }$ is a Banach function space endowed with the norm
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040103.png" /></td> </tr></table>
+
\begin{equation*} \| x ^ { \prime } \| _ { X ^ { \prime } } = \operatorname { sup } \{ \int _ { \Omega } | x x ^ { \prime } | d \mu : \| x \| _ { X } \leq 1 \}. \end{equation*}
  
Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040104.png" /> isometrically if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040105.png" /> has the Fatou property.
+
Moreover, $X ^ { \prime \prime } = X$ isometrically if and only if $X$ has the Fatou property.
  
It is important to describe the relation between the Köthe dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040106.png" /> and the usual (topological) dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040107.png" /> of a Banach function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040108.png" />. A [[Linear functional|linear functional]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040109.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040110.png" /> is said to be order continuous (or integral) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040111.png" /> for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040112.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040113.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040114.png" /> almost everywhere. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040115.png" /> be the space all order-continuous functionals. This is a closed and norm-one complemented subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040116.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040117.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040118.png" /> denotes a complement to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040119.png" />, called the space of all singular functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040120.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040121.png" /> is always total on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040122.png" /> (cf. [[Total set|Total set]]). Furthermore, it is norming, i.e.,
+
It is important to describe the relation between the Köthe dual $X ^ { \prime }$ and the usual (topological) dual space $X ^ { * }$ of a Banach function space $X$. A [[Linear functional|linear functional]] $f$ on $X$ is said to be order continuous (or integral) if $f ( x _ { n } ) \rightarrow 0$ for every sequence $\{ x _ { n } \}$ in $X$ such that $x _ { n } \downarrow 0$ almost everywhere. Let $X ^ { * }_{c}$ be the space all order-continuous functionals. This is a closed and norm-one complemented subspace of $X ^ { * }$. Thus, $X ^ { * } = X _ { c } ^ { * } \oplus X _ { s } ^ { * }$, where $X _ { s } ^ { * }$ denotes a complement to $X ^ { * }_{c}$, called the space of all singular functionals on $X$. The space $X ^ { * }_{c}$ is always total on $X$ (cf. [[Total set|Total set]]). Furthermore, it is norming, i.e.,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040123.png" /></td> </tr></table>
+
\begin{equation*} \| x \| _ { X } = \operatorname { sup } \left\{ \left| \int _ { \Omega } x x ^ { \prime } d \mu \right| : x ^ { \prime } \in X ^ { \prime } , \| x ^ { \prime } \| _ { X ^ { \prime } } \leq 1 \right\}, \end{equation*}
  
if and only if the norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040124.png" /> is order semi-continuous, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040125.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040126.png" />. The mapping that assigns to every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040127.png" /> the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040128.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040129.png" /> is an order-linear isometry from the Köthe dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040130.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040131.png" />. In this way <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040132.png" /> is identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040133.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040134.png" /> is an order-continuous Banach function space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040135.png" /> can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040136.png" /> (see [[#References|[a10]]], [[#References|[a12]]], [[#References|[a14]]]).
+
if and only if the norm on $X$ is order semi-continuous, i.e., $\| f _ { n } \| \rightarrow \| f \|$ whenever $0 \leq f _ { n } \uparrow f \in X$. The mapping that assigns to every $x ^ { \prime } \in X ^ { \prime }$ the functional $x \mapsto \int _ { \Omega } x x ^ { \prime } d \mu$ on $X$ is an order-linear isometry from the Köthe dual space $X ^ { \prime }$ onto $X ^ { * }_{c}$. In this way $X ^ { * }_{c}$ is identified with $X ^ { \prime }$. In particular, if $X$ is an order-continuous Banach function space, then $X ^ { * }$ can be identified with $X ^ { \prime }$ (see [[#References|[a10]]], [[#References|[a12]]], [[#References|[a14]]]).
  
There are many methods of constructing Banach function spaces which are intermediate in some sense between two given Banach function spaces. One such method is the following construction, again due to Calderón [[#References|[a5]]]. See also [[#References|[a13]]] for the generalized version due to G.A. Lozanovskii. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040137.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040138.png" /> be two Banach function spaces on the same measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040139.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040140.png" />, the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040141.png" /> is defined to be the space of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040142.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040143.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040144.png" />-almost everywhere for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040145.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040146.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040147.png" /> is a Banach function space endowed with the norm
+
There are many methods of constructing Banach function spaces which are intermediate in some sense between two given Banach function spaces. One such method is the following construction, again due to Calderón [[#References|[a5]]]. See also [[#References|[a13]]] for the generalized version due to G.A. Lozanovskii. Let $X _ { 0 }$ and $X _ { 1 }$ be two Banach function spaces on the same measure space $( \Omega , \Sigma , \mu )$. For each $\theta \in ( 0,1 )$, the lattice $X _ { \theta } = X _ { 0 } ^ { 1 - \theta } X _ { 1 } ^ { \theta }$ is defined to be the space of all $x \in L ^ { 0 } ( \mu )$ such that $| x | = | x _ { 0 } | ^ { 1 - \theta } | x _ { 1 } | ^ { \theta }$ $\mu$-almost everywhere for some $x _ { 0 } \in X _ { 0 }$ and $x _ { 1 } \in X _ { 1 }$. The space $X _ { \theta }$ is a Banach function space endowed with the norm
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040148.png" /></td> </tr></table>
+
\begin{equation*} \| x  \| _ { \theta } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040149.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040149.png"/></td> </tr></table>
  
The identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040150.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040151.png" /> is an important result proved by Lozanovskii [[#References|[a13]]]. Closely related results are the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040152.png" />, which holds for any Banach function space on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040153.png" />, and also the Lozanovskii factorization theorem: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040154.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040155.png" /> there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040156.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040157.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040158.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040159.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040160.png" /> has the Fatou property, the theorem is true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040161.png" /> as well.
+
The identity $( X _ { 0 } ^ { 1 - \theta } X _ { 1 } ^ { \theta } ) ^ { \prime } = ( X _ { 0 } ^ { \prime } ) ^ { 1 - \theta } ( X _ { 1 } ^ { \prime } ) ^ { \theta }$ for all $0 &lt; \theta &lt; 1$ is an important result proved by Lozanovskii [[#References|[a13]]]. Closely related results are the formula $X ^ { 1 / 2 } ( X ^ { \prime } ) ^ { 1 / 2 } = L _ { 2 }$, which holds for any Banach function space on $( \Omega , \Sigma , \mu )$, and also the Lozanovskii factorization theorem: For every $f \in L _ { 1 } ( \mu )$ and $\epsilon &gt; 0$ there exist $x \in X$ and $y \in X ^ { \prime }$ such that $f = x y$ and $\|x \|_X \| y  \| _ { X ^ { \prime } } \leq ( 1 + \epsilon ) \| f \| _ { L _ { 1 } }$. If $X$ has the Fatou property, the theorem is true for $\epsilon = 0$ as well.
  
 
This theorem has proved to be very useful in various applications (see, for example, [[#References|[a9]]], [[#References|[a15]]]).
 
This theorem has proved to be very useful in various applications (see, for example, [[#References|[a9]]], [[#References|[a15]]]).
  
Calderón's construction has found many other interesting applications in the study of Banach function spaces. An example is Pisier's theorem [[#References|[a16]]], which says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040162.png" />, then a Banach function space on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040163.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040166.png" />-convex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040167.png" />-concave if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040168.png" /> for some Banach function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040169.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040170.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040171.png" />. An application of this result and interpolation yields the following (see [[#References|[a16]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040172.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040173.png" />-convex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040174.png" />-concave Banach function space for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040175.png" />. Then every bounded linear operator from an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040176.png" />-space into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040177.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040179.png" />-summing with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040180.png" />, i.e., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040181.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040182.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040183.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040184.png" /> (cf. also [[Absolutely summing operator|Absolutely summing operator]]).
+
Calderón's construction has found many other interesting applications in the study of Banach function spaces. An example is Pisier's theorem [[#References|[a16]]], which says that if $1 &lt; p &lt; 2$, then a Banach function space on $( \Omega , \Sigma , \mu )$ is $p$-convex and $p ^ { \prime }$-concave if and only if $X = ( X _ { 0 } ) ^ { 1 - \theta } ( L _ { 2 } ( \mu ) ) ^ { \theta }$ for some Banach function space $X _ { 0 }$ on $( \Omega , \Sigma , \mu )$, with $\theta = 1 - 1 / p = 1 / p ^ { \prime }$. An application of this result and interpolation yields the following (see [[#References|[a16]]]): Let $X$ be a $p$-convex and $p ^ { \prime }$-concave Banach function space for some $1 &lt; p &lt; 2$. Then every bounded linear operator from an $L_1$-space into $X$ is $( r , 1 )$-summing with $1 / r = 1 / p ^ { \prime } + 1 / 2$, i.e., if $( x _ { n } ) \subset L _ { 1 }$ is such that $\Sigma _ { n = 1 } ^ { \infty } | x ^ { * } ( x _ { n } ) | &lt; \infty$ for all $x ^ { * } \in L _ { \infty }$, then $\Sigma _ { n = 1 } ^ { \infty } \| T _ { x _ { n } } \| _ { X } ^ { r } &lt; \infty$ (cf. also [[Absolutely summing operator|Absolutely summing operator]]).
  
For another example see [[#References|[a6]]], where the Calderón construction is used to construct a class of super-reflexive and complementably minimal Banach spaces (i.e., such that every infinite-dimensional closed subspace contains a complemented subspace isomorphic to a given space of this class) which are not isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040185.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040186.png" />.
+
For another example see [[#References|[a6]]], where the Calderón construction is used to construct a class of super-reflexive and complementably minimal Banach spaces (i.e., such that every infinite-dimensional closed subspace contains a complemented subspace isomorphic to a given space of this class) which are not isomorphic to $\text{I} _ { p }$ for any $p$.
  
One of the most interesting problems in the theory of Banach function spaces is to determine when two Banach function spaces which are isomorphic as Banach spaces are also lattice isomorphic. The first result of this type, due to Y.A. Abramovich and P. Wojtaszczyk [[#References|[a1]]] says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040187.png" /> has a unique structure as a non-atomic Banach function space (i.e., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040188.png" /> is a non-atomic Banach function space isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040189.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040190.png" /> is lattice isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040191.png" />). The general study of possible rearrangement-invariant lattice structures in in Banach function spaces on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040192.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040193.png" /> was initiated in [[#References|[a7]]], where, among other important results, it is shown that any rearrangement-invariant function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040194.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040195.png" /> which is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040196.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040197.png" />, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040198.png" /> up to an equivalent renorming. See also [[#References|[a8]]], where important general results on the uniqueness of the structure of Banach function spaces are presented.
+
One of the most interesting problems in the theory of Banach function spaces is to determine when two Banach function spaces which are isomorphic as Banach spaces are also lattice isomorphic. The first result of this type, due to Y.A. Abramovich and P. Wojtaszczyk [[#References|[a1]]] says that $L_1$ has a unique structure as a non-atomic Banach function space (i.e., if $X$ is a non-atomic Banach function space isomorphic to $L_1$, then $X$ is lattice isomorphic to $L_1$). The general study of possible rearrangement-invariant lattice structures in in Banach function spaces on $[0,1]$ or $[ 0 , \infty )$ was initiated in [[#References|[a7]]], where, among other important results, it is shown that any rearrangement-invariant function space $X$ on $[0,1]$ which is isomorphic to $L _ { p } ( 0,1 )$, $1 \leq p \leq \infty$, is equal to $L _ { p } ( 0,1 )$ up to an equivalent renorming. See also [[#References|[a8]]], where important general results on the uniqueness of the structure of Banach function spaces are presented.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y.A. Abramovich,  P. Wojtaszczyk,  "On the uniqueness of order in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040199.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040200.png" />"  ''Mat. Zametki'' , '''18'''  (1975)  pp. 313–325</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.D. Aliprantis,  O. Burkinshaw,  "Positive operators" , Acad. Press  (1995)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Alspach,  P. Enflo,  E. Odell,  "On the structure of separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040201.png" /> spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120040/b120040202.png" />"  ''Studia Math.'' , '''60'''  (1977)  pp. 79–90</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Bennett,  R. Sharpley,  "Interpolation of operators" , Acad. Press  (1988)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.P. Calderón,  "Intermediate spaces and interpolation, the complex method"  ''Studia Math.'' , '''24'''  (1964)  pp. 113–190</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.G. Casazza,  N.J. Kalton,  D. Kutzarova,  M. Mastylo,  "Complex interpolation and complementably minimal spaces"  N. Kalton (ed.)  E. Saab (ed.)  S. Montgomery-Smith (ed.) , ''Interaction between Functional Analysis, Harmonic Analysis, and Probability (Proc. Conf. Univ. Missouri 1994)'' , ''Lecture Notes Pure Appl. Math.'' , '''175''' , M. Dekker  (1996)  pp. 135–143</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W.B. Johnson,  B. Maurey, V. Schechtmannn,  L. Tzafriri,  "Symmetric structures in Banach spaces"  ''Memoirs Amer. Math. Soc.'' , '''217'''  (1979)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  N.J. Kalton,  "Lattice structures on Banach spaces"  ''Memoirs Amer. Math. Soc.'' , '''493'''  (1993)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  N.J. Kalton,  "The basic sequence problem"  ''Studia Math.'' , '''116'''  (1995)  pp. 167–187</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis" , Pergamon  (1998)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (In Russian)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J. Lindenstrauss,  L. Tzafriri,  "Classical Banach spaces: Function spaces" , '''2''' , Springer  (1979)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  G.A. Lozanovskii,  "On some Banach lattices"  ''Sib. Math. J.'' , '''10'''  (1969)  pp. 419–430</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''2''' , North-Holland  (1983)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  E. Odell,  T. Schlumprecht,  "The distortion problem"  ''Acta Math.'' , '''173'''  (1994)  pp. 258–281</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  G. Pisier,  "Some applications of the complex interpolation method to Banach lattices"  ''J. Anal. Math.'' , '''35'''  (1979)  pp. 264–281</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  Y.A. Abramovich,  P. Wojtaszczyk,  "On the uniqueness of order in the spaces $\text{I} _ { p }$ and $L _ { p } [ 0,1 ]$"  ''Mat. Zametki'' , '''18'''  (1975)  pp. 313–325</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  C.D. Aliprantis,  O. Burkinshaw,  "Positive operators" , Acad. Press  (1995)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D. Alspach,  P. Enflo,  E. Odell,  "On the structure of separable $\mathcal{L} _ { p }$ spaces, $( 1 &lt; p &lt; \infty )$"  ''Studia Math.'' , '''60'''  (1977)  pp. 79–90</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C. Bennett,  R. Sharpley,  "Interpolation of operators" , Acad. Press  (1988)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A.P. Calderón,  "Intermediate spaces and interpolation, the complex method"  ''Studia Math.'' , '''24'''  (1964)  pp. 113–190</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  P.G. Casazza,  N.J. Kalton,  D. Kutzarova,  M. Mastylo,  "Complex interpolation and complementably minimal spaces"  N. Kalton (ed.)  E. Saab (ed.)  S. Montgomery-Smith (ed.) , ''Interaction between Functional Analysis, Harmonic Analysis, and Probability (Proc. Conf. Univ. Missouri 1994)'' , ''Lecture Notes Pure Appl. Math.'' , '''175''' , M. Dekker  (1996)  pp. 135–143</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  W.B. Johnson,  B. Maurey, V. Schechtmannn,  L. Tzafriri,  "Symmetric structures in Banach spaces"  ''Memoirs Amer. Math. Soc.'' , '''217'''  (1979)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  N.J. Kalton,  "Lattice structures on Banach spaces"  ''Memoirs Amer. Math. Soc.'' , '''493'''  (1993)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  N.J. Kalton,  "The basic sequence problem"  ''Studia Math.'' , '''116'''  (1995)  pp. 167–187</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis" , Pergamon  (1998)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (In Russian)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  J. Lindenstrauss,  L. Tzafriri,  "Classical Banach spaces: Function spaces" , '''2''' , Springer  (1979)</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  G.A. Lozanovskii,  "On some Banach lattices"  ''Sib. Math. J.'' , '''10'''  (1969)  pp. 419–430</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''2''' , North-Holland  (1983)</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  E. Odell,  T. Schlumprecht,  "The distortion problem"  ''Acta Math.'' , '''173'''  (1994)  pp. 258–281</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  G. Pisier,  "Some applications of the complex interpolation method to Banach lattices"  ''J. Anal. Math.'' , '''35'''  (1979)  pp. 264–281</td></tr></table>

Revision as of 17:02, 1 July 2020

Let $( \Omega , \Sigma , \mu )$ be a complete $\sigma$-finite measure space and let $L ^ { 0 } ( \mu ) = L ^ { 0 } ( \Omega , \Sigma , \mu )$ be the space of all equivalence classes of $\mu$-measurable real-valued functions endowed with the topology of convergence in measure relative to each set of finite measure.

A Banach space $X \subset L ^ { 0 } ( \mu )$ is called a Banach function space on $( \Omega , \Sigma , \mu )$ if there exists a $u \in X$ such that $u > 0$ almost everywhere and $X$ satisfies the ideal property:

\begin{equation*} x \in L ^ { 0 } ( \mu ) , y \in X , | x | \leq | y | \mu - a.e. \end{equation*}

\begin{equation*} \Downarrow x \in X \text { and } \| x \| \leq \| y \|. \end{equation*}

The Lebesgue function spaces $L _ { p }$ ($1 \leq p \leq \infty$) play a primary role in many problems arising in mathematical analysis. There are other classes of Banach function spaces that are also of interest. The classes of Musielak–Orlicz, Lorentz and Marcinkiewicz spaces, for example, are of intrinsic importance (cf. also Orlicz space; Orlicz–Lorentz space; Marcinkiewicz space). Function spaces are important and natural examples of abstract Banach lattices (a Banach lattice is a Banach space that is also a vector lattice $X$ with the property that $\| x \| \leq \| y \|$ whenever $| x | \leq | y |$, where $| x | = x \vee ( - x )$, cf. also Banach lattice). A Banach lattice is said to be order continuous if $\| x _ { n } \| \rightarrow 0$ whenever $x _ { n } \downarrow 0$. The following very useful general representation result (see [a12]) allows one to reduce most of the proofs for a quite large class of abstract Banach lattices to the case of Banach function spaces: Let $X$ be an order-continuous Banach lattice with a weak unit (a weak unit is an element $e > 0$ such that $e \wedge | x | = 0$ implies $x = 0$). Then there exist a probability space $( \Omega , \Sigma , \mu )$ and a Banach function space $X$ on $( \Omega , \Sigma , \mu )$ such that $X$ is isometrically lattice-isomorphic to $X$ and $L _ { \infty } ( \mu ) \subset X \subset L _ { 1 } ( \mu )$ with continuous inclusions.

See [a2], [a7], [a10], [a14] for a general theory of Banach lattices.

A Banach function space $X$ is said to have the Fatou property if whenever $( f _ { n } )$ is a norm-bounded sequence in $X$ such that $0 \leq f _ { n } \uparrow f \in L ^ { 0 } ( \mu )$, then $f \in X$ and $\| f _ { n } \| \rightarrow \| f \|$.

In recent (1998) years a great deal of research went into the study of rearrangement-invariant function spaces, in particular of Orlicz spaces. General references to this area are e.g. [a7], [a11], [a12]. A Banach function space $X$ is said to be rearrangement invariant if whenever $f \in X$, $g \in L ^ { 0 } ( \mu )$, and $f$ and $g$ are equi-measurable, then $g \in X$ and $\| f \| = \| g \|$. Two functions $f$ and $g$ are called equi-measurable if $| f|$ and $| g |$ have identical distributions, that is,

\begin{equation*} \mu _ { f } ( \lambda ) = \mu \{ t \in \Omega : | f ( t ) | > \lambda \} = \mu _ { g } ( \lambda ) \end{equation*}

for all $\lambda > 0$.

In the study of rearrangement-invariant function spaces, the Boyd indices play an important role (see e.g. [a7], [a12], and Boyd index). The Boyd indices $p_{X} $ and $q_X$ of a rearrangement-invariant function space $X$ on $[0,1]$ or $[ 0 , \infty )$ are defined by

\begin{equation*} p _ { X } = \operatorname { lim } _ { s \rightarrow \infty } \frac { \operatorname { log } s } { \operatorname { log } \| D _ { s } \| _ { X } }, \end{equation*}

\begin{equation*} q _ { X } = \operatorname { lim } _ { s \rightarrow 0 + } \frac { \operatorname { log } s } { \operatorname { log } \| D _ { s } \| _ { X } }, \end{equation*}

where for $s > 0$, $D _ { S }$ denotes the dilation operator, defined by $D _ { s } f ( t ) = f ( t / s )$ for $f \in X$ (where $f$ is defined to be zero outside $[0,1]$ in the former case).

For example, consider the following results, which hold for every separable rearrangement-invariant function space $X$ on $[0,1]$:

i) $X$ has an unconditional basis if and only if $1 < p_{ X}$ and $q_{X} < \infty$ (see, e.g., [a11], [a12]);

ii) if $1 < p_{ X}$ and $q_{X} < \infty$, then $X$ is a primary, i.e., whenever $X = E \oplus F$, then at least one of $E$ and $F$ is isomorphic to $X$ (see [a3]).

Rearrangement-invariant function spaces play an important role in the theory of interpolation of operators (see [a4], [a11]). A remarkable result of A.P. Calderón [a5] on the characterization of all interpolation spaces between $L _ { 1 } = L _ { 1 } ( \mu )$ and $L _ { \infty } = L _ { \infty } ( \mu )$ asserts that $X$ is an interpolation space with respect to the couple $( L _ { 1 } , L _ { \infty } )$ (i.e., that every linear operator $T : L _ { 1 } + L _ { \infty } \rightarrow L _ { 1 } + L _ { \infty }$ such that $T : L _ { 1 } \rightarrow L _ { 1 }$ and $T : L _ { \infty } \rightarrow L _ { \infty }$ boundedly, also maps $X$ to $X$ boundedly) if and only if it has the following property: For every $g \in X$ and every $f \in L _ { 1 } + L _ { \infty }$, whenever $\int _ { 0 } ^ { t } f ^ { * } ( s ) d s \leq \int _ { 0 } ^ { t } g ^ { * } ( s ) d s$ for all $t > 0$, it follows that $f \in X$ and $\| f \|_X \leq C\| g \|_X$ for some absolute constant $C$.

Here, $f ^ { * }$ denotes the non-increasing rearrangement of $f$, which is defined by

\begin{equation*} f ^ { * } ( t ) = \operatorname { inf } \{ \lambda > 0 : \mu _ { f } ( \lambda ) \leq t \} \end{equation*}

for $t > 0$. In particular, Calderón's result implies that rearrangement-invariant function spaces which have the Fatou property or are separable are interpolation spaces between $L_1$ and $L _ { \infty }$.

The Köthe dual space $X ^ { \prime }$ of a Banach function space $X$ on $( \Omega , \Sigma , \mu )$ is defined to be the space of all $x ^ { \prime }$ for which $x x ^ { \prime } \in L _ { 1 } ( \mu )$ for each $x \in X$ (cf. also Köthe–Toeplitz dual). The space $X ^ { \prime }$ is a Banach function space endowed with the norm

\begin{equation*} \| x ^ { \prime } \| _ { X ^ { \prime } } = \operatorname { sup } \{ \int _ { \Omega } | x x ^ { \prime } | d \mu : \| x \| _ { X } \leq 1 \}. \end{equation*}

Moreover, $X ^ { \prime \prime } = X$ isometrically if and only if $X$ has the Fatou property.

It is important to describe the relation between the Köthe dual $X ^ { \prime }$ and the usual (topological) dual space $X ^ { * }$ of a Banach function space $X$. A linear functional $f$ on $X$ is said to be order continuous (or integral) if $f ( x _ { n } ) \rightarrow 0$ for every sequence $\{ x _ { n } \}$ in $X$ such that $x _ { n } \downarrow 0$ almost everywhere. Let $X ^ { * }_{c}$ be the space all order-continuous functionals. This is a closed and norm-one complemented subspace of $X ^ { * }$. Thus, $X ^ { * } = X _ { c } ^ { * } \oplus X _ { s } ^ { * }$, where $X _ { s } ^ { * }$ denotes a complement to $X ^ { * }_{c}$, called the space of all singular functionals on $X$. The space $X ^ { * }_{c}$ is always total on $X$ (cf. Total set). Furthermore, it is norming, i.e.,

\begin{equation*} \| x \| _ { X } = \operatorname { sup } \left\{ \left| \int _ { \Omega } x x ^ { \prime } d \mu \right| : x ^ { \prime } \in X ^ { \prime } , \| x ^ { \prime } \| _ { X ^ { \prime } } \leq 1 \right\}, \end{equation*}

if and only if the norm on $X$ is order semi-continuous, i.e., $\| f _ { n } \| \rightarrow \| f \|$ whenever $0 \leq f _ { n } \uparrow f \in X$. The mapping that assigns to every $x ^ { \prime } \in X ^ { \prime }$ the functional $x \mapsto \int _ { \Omega } x x ^ { \prime } d \mu$ on $X$ is an order-linear isometry from the Köthe dual space $X ^ { \prime }$ onto $X ^ { * }_{c}$. In this way $X ^ { * }_{c}$ is identified with $X ^ { \prime }$. In particular, if $X$ is an order-continuous Banach function space, then $X ^ { * }$ can be identified with $X ^ { \prime }$ (see [a10], [a12], [a14]).

There are many methods of constructing Banach function spaces which are intermediate in some sense between two given Banach function spaces. One such method is the following construction, again due to Calderón [a5]. See also [a13] for the generalized version due to G.A. Lozanovskii. Let $X _ { 0 }$ and $X _ { 1 }$ be two Banach function spaces on the same measure space $( \Omega , \Sigma , \mu )$. For each $\theta \in ( 0,1 )$, the lattice $X _ { \theta } = X _ { 0 } ^ { 1 - \theta } X _ { 1 } ^ { \theta }$ is defined to be the space of all $x \in L ^ { 0 } ( \mu )$ such that $| x | = | x _ { 0 } | ^ { 1 - \theta } | x _ { 1 } | ^ { \theta }$ $\mu$-almost everywhere for some $x _ { 0 } \in X _ { 0 }$ and $x _ { 1 } \in X _ { 1 }$. The space $X _ { \theta }$ is a Banach function space endowed with the norm

\begin{equation*} \| x \| _ { \theta } = \end{equation*}

The identity $( X _ { 0 } ^ { 1 - \theta } X _ { 1 } ^ { \theta } ) ^ { \prime } = ( X _ { 0 } ^ { \prime } ) ^ { 1 - \theta } ( X _ { 1 } ^ { \prime } ) ^ { \theta }$ for all $0 < \theta < 1$ is an important result proved by Lozanovskii [a13]. Closely related results are the formula $X ^ { 1 / 2 } ( X ^ { \prime } ) ^ { 1 / 2 } = L _ { 2 }$, which holds for any Banach function space on $( \Omega , \Sigma , \mu )$, and also the Lozanovskii factorization theorem: For every $f \in L _ { 1 } ( \mu )$ and $\epsilon > 0$ there exist $x \in X$ and $y \in X ^ { \prime }$ such that $f = x y$ and $\|x \|_X \| y \| _ { X ^ { \prime } } \leq ( 1 + \epsilon ) \| f \| _ { L _ { 1 } }$. If $X$ has the Fatou property, the theorem is true for $\epsilon = 0$ as well.

This theorem has proved to be very useful in various applications (see, for example, [a9], [a15]).

Calderón's construction has found many other interesting applications in the study of Banach function spaces. An example is Pisier's theorem [a16], which says that if $1 < p < 2$, then a Banach function space on $( \Omega , \Sigma , \mu )$ is $p$-convex and $p ^ { \prime }$-concave if and only if $X = ( X _ { 0 } ) ^ { 1 - \theta } ( L _ { 2 } ( \mu ) ) ^ { \theta }$ for some Banach function space $X _ { 0 }$ on $( \Omega , \Sigma , \mu )$, with $\theta = 1 - 1 / p = 1 / p ^ { \prime }$. An application of this result and interpolation yields the following (see [a16]): Let $X$ be a $p$-convex and $p ^ { \prime }$-concave Banach function space for some $1 < p < 2$. Then every bounded linear operator from an $L_1$-space into $X$ is $( r , 1 )$-summing with $1 / r = 1 / p ^ { \prime } + 1 / 2$, i.e., if $( x _ { n } ) \subset L _ { 1 }$ is such that $\Sigma _ { n = 1 } ^ { \infty } | x ^ { * } ( x _ { n } ) | < \infty$ for all $x ^ { * } \in L _ { \infty }$, then $\Sigma _ { n = 1 } ^ { \infty } \| T _ { x _ { n } } \| _ { X } ^ { r } < \infty$ (cf. also Absolutely summing operator).

For another example see [a6], where the Calderón construction is used to construct a class of super-reflexive and complementably minimal Banach spaces (i.e., such that every infinite-dimensional closed subspace contains a complemented subspace isomorphic to a given space of this class) which are not isomorphic to $\text{I} _ { p }$ for any $p$.

One of the most interesting problems in the theory of Banach function spaces is to determine when two Banach function spaces which are isomorphic as Banach spaces are also lattice isomorphic. The first result of this type, due to Y.A. Abramovich and P. Wojtaszczyk [a1] says that $L_1$ has a unique structure as a non-atomic Banach function space (i.e., if $X$ is a non-atomic Banach function space isomorphic to $L_1$, then $X$ is lattice isomorphic to $L_1$). The general study of possible rearrangement-invariant lattice structures in in Banach function spaces on $[0,1]$ or $[ 0 , \infty )$ was initiated in [a7], where, among other important results, it is shown that any rearrangement-invariant function space $X$ on $[0,1]$ which is isomorphic to $L _ { p } ( 0,1 )$, $1 \leq p \leq \infty$, is equal to $L _ { p } ( 0,1 )$ up to an equivalent renorming. See also [a8], where important general results on the uniqueness of the structure of Banach function spaces are presented.

References

[a1] Y.A. Abramovich, P. Wojtaszczyk, "On the uniqueness of order in the spaces $\text{I} _ { p }$ and $L _ { p } [ 0,1 ]$" Mat. Zametki , 18 (1975) pp. 313–325
[a2] C.D. Aliprantis, O. Burkinshaw, "Positive operators" , Acad. Press (1995)
[a3] D. Alspach, P. Enflo, E. Odell, "On the structure of separable $\mathcal{L} _ { p }$ spaces, $( 1 < p < \infty )$" Studia Math. , 60 (1977) pp. 79–90
[a4] C. Bennett, R. Sharpley, "Interpolation of operators" , Acad. Press (1988)
[a5] A.P. Calderón, "Intermediate spaces and interpolation, the complex method" Studia Math. , 24 (1964) pp. 113–190
[a6] P.G. Casazza, N.J. Kalton, D. Kutzarova, M. Mastylo, "Complex interpolation and complementably minimal spaces" N. Kalton (ed.) E. Saab (ed.) S. Montgomery-Smith (ed.) , Interaction between Functional Analysis, Harmonic Analysis, and Probability (Proc. Conf. Univ. Missouri 1994) , Lecture Notes Pure Appl. Math. , 175 , M. Dekker (1996) pp. 135–143
[a7] W.B. Johnson, B. Maurey, V. Schechtmannn, L. Tzafriri, "Symmetric structures in Banach spaces" Memoirs Amer. Math. Soc. , 217 (1979)
[a8] N.J. Kalton, "Lattice structures on Banach spaces" Memoirs Amer. Math. Soc. , 493 (1993)
[a9] N.J. Kalton, "The basic sequence problem" Studia Math. , 116 (1995) pp. 167–187
[a10] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1998)
[a11] S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (In Russian)
[a12] J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces: Function spaces" , 2 , Springer (1979)
[a13] G.A. Lozanovskii, "On some Banach lattices" Sib. Math. J. , 10 (1969) pp. 419–430
[a14] W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , 2 , North-Holland (1983)
[a15] E. Odell, T. Schlumprecht, "The distortion problem" Acta Math. , 173 (1994) pp. 258–281
[a16] G. Pisier, "Some applications of the complex interpolation method to Banach lattices" J. Anal. Math. , 35 (1979) pp. 264–281
How to Cite This Entry:
Banach function space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach_function_space&oldid=50444
This article was adapted from an original article by M. Mastyło (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article