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Consider the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b1203201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b1203202.png" />, and a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b1203203.png" />. Since the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b1203204.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b1203205.png" />-additive, it is easily seen that the following condition is satisfied: For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b1203206.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b1203207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b1203208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b1203209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032010.png" /> (in the sense of disjoint support), one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032011.png" />.
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In [[#References|[a2]]] H.F. Bohnenblust showed that the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032012.png" /> are the only Banach lattices (cf. also [[Banach lattice|Banach lattice]]) possessing this property; more precisely, he proved the following theorem, now known as the Bohnenblust theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032013.png" /> be a Banach lattice of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032014.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032020.png" />. Then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032022.png" />, such that the norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032023.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032024.png" />-additive.
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Out of 107 formulas, 106 were replaced by TEX code.-->
  
Here, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032025.png" />, a norm is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032027.png" />-additive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032028.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032029.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032030.png" />; a norm is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032032.png" />-additive, or, equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032033.png" /> is said to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032035.png" />-space, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032036.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032037.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032038.png" />.
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Consider the space $L ^ { p } ( \mu )$, $1 \leq p \leq \infty$, and a [[Measure space|measure space]] $( \Omega , \mathcal A , \mu )$. Since the norm $\| \cdot \| p$ is $p$-additive, it is easily seen that the following condition is satisfied: For all $x , y , u , v \in L ^ { P } ( \mu )$ satisfying $\| x  \| _ { p } = \| u  \| _ { p }$, $\| y \| _ { p } = \| v \| _ { p }$, $x \perp y$ and $u \perp v$ (in the sense of disjoint support), one has $\| x + y \| _ { p } = \| u + v \| _ { p }$.
  
It should be noted that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032039.png" />, every Banach lattice with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032040.png" />-additive norm is isometrically isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032041.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032042.png" /> a suitable measure space. This representation theorem is essentially due to S. Kakutani [[#References|[a3]]], who considered the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032043.png" />; the proof of the more general result follows almost the same lines. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032044.png" /> the situation is not so clear: there exist many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032045.png" />-spaces that are not isomorphic to any concrete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032046.png" />-space, for instance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032047.png" />.
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In [[#References|[a2]]] H.F. Bohnenblust showed that the spaces $L ^ { p } ( \mu )$ are the only Banach lattices (cf. also [[Banach lattice|Banach lattice]]) possessing this property; more precisely, he proved the following theorem, now known as the Bohnenblust theorem: Let $E$ be a Banach lattice of dimension $\geq 3$ satisfying $\| x + y \| = \| u + v \|$ for all $x , y , u , v \in E$ such that $\| x \|  = \| u \|$, $\| y \| = \| v \|$, $x \perp y$, and $u \perp v$. Then there exists a $p$, $1 \leq p \leq \infty$, such that the norm on $E$ is $p$-additive.
  
In the proof of his theorem, H.F. Bohnenblust introduced an interesting and tricky method to construct the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032048.png" /> such that the norm is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032049.png" />-additive. A similar method was used later by M. Zippin [[#References|[a5]]] to characterize <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032051.png" />-spaces in terms of bases. Since the proof given by Bohnenblust is interesting in itself, the main ideas are sketched below.
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Here, for $p < \infty$, a norm is said to be $p$-additive if $\| x \| ^ { p } + \| y \| ^ { p } = \| x + y \| ^ { p }$ for all $x , y \in E$ with $x \perp y$; a norm is said to be $\infty$-additive, or, equivalently, $E$ is said to be an $M$-space, if $\| x + y \| = \operatorname { max } \{ \| x \| , \| y \| \}$ for all $x , y \in E$ with $x \perp y$.
  
By hypothesis, there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032052.png" /> defined by
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It should be noted that when $1 \leq p < \infty$, every Banach lattice with a $p$-additive norm is isometrically isomorphic to $L ^ { p } ( \mu )$, with $( \Omega , \mathcal A , \mu )$ a suitable measure space. This representation theorem is essentially due to S. Kakutani [[#References|[a3]]], who considered the case $p = 1$; the proof of the more general result follows almost the same lines. For $p = \infty$ the situation is not so clear: there exist many $M$-spaces that are not isomorphic to any concrete $L ^ { \infty } ( \mu )$-space, for instance $c_0$.
  
<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/b120320/b12032053.png" /></td> </tr></table>
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In the proof of his theorem, H.F. Bohnenblust introduced an interesting and tricky method to construct the $p$ such that the norm is $p$-additive. A similar method was used later by M. Zippin [[#References|[a5]]] to characterize $\mathbf{l}^{p}$-spaces in terms of bases. Since the proof given by Bohnenblust is interesting in itself, the main ideas are sketched below.
  
whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032055.png" /> are disjoint vectors of norm one. It can easily be verified that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032056.png" /> has the following properties:
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By hypothesis, there exists a function $F : [ 0 , \infty ) ^ { 2 } \rightarrow [ 0 , \infty )$ defined by
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032057.png" />;
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\begin{equation*} F ( s , t ) = \| t x + s y \| \text { for all } s , t \geq 0, \end{equation*}
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032058.png" />;
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whenever $x$ and $y$ are disjoint vectors of norm one. It can easily be verified that the function $F$ has the following properties:
  
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032059.png" />;
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1) $F ( 0 , t ) = t$;
  
iv) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032060.png" />;
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ii) $F ( s , t ) = F ( t , s )$;
  
v) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032061.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032063.png" />. The only non-trivial inclusion iv) follows from
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iii) $F ( r s , r t ) = r F ( s , t )$;
  
<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/b120320/b12032064.png" /></td> </tr></table>
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iv) $F ( r , F ( s , t ) ) = F ( F ( r , s ) , t )$;
  
<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/b120320/b12032065.png" /></td> </tr></table>
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v) $F ( s , t ) \leq F ( s _ { 1 } , t _ { 1 } )$ if $s\leq s_ 1$, $t \leq t_1$. The only non-trivial inclusion iv) follows from
  
<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/b120320/b12032066.png" /></td> </tr></table>
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\begin{equation*} F ( r , F ( s , t ) ) = \| r x + \| s y + t z \| z \| = \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/b120320/b12032067.png" /></td> </tr></table>
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\begin{equation*} = F ( s , t ) \left\| \frac { r } { F ( s , t ) } x + z \right\| = \end{equation*}
  
for all disjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032068.png" /> of norm one and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032069.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032070.png" />.
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\begin{equation*} = F ( s , t ) \left\| \frac { r } { F ( s , t ) } x + \frac { 1 } { F ( s , t ) } ( s y + t z ) \right\| = \end{equation*}
  
One defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032072.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032073.png" />. Property v) implies that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032074.png" /> is increasing. By induction one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032076.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032077.png" />, then properties i)–v) easily imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032078.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032079.png" />. Hence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032080.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032081.png" />-space.
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\begin{equation*} = \| r x + s y + t z \| = F ( F ( r , s ) , t ) \end{equation*}
  
Assume now that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032082.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032083.png" />. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032084.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032085.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032086.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032087.png" /> is an increasing sequence, one concludes from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032088.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032089.png" /> or, equivalently, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032090.png" />. This yields
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for all disjoint $x , y , z \in E _ { + }$ of norm one and $r , s , t \geq 0$ with $t \neq 0$.
  
<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/b120320/b12032091.png" /></td> </tr></table>
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One defines $a _ { 1 } = 1$ and $a _ { n + 1} = F ( 1 , a _ { n } )$ for all $n \geq 1$. Property v) implies that the sequence $( a _ { n } ) _ { n = 1 } ^ { \infty }$ is increasing. By induction one obtains $a _ { n  + m} = F ( a _ { n } , a _ { m } )$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032076.png"/>. If $a _ { 2 } = 1$, then properties i)–v) easily imply $F ( s , t ) = \operatorname { max } \{ s , t \}$ for all $s , t \geq 0$. Hence, $E$ is an $M$-space.
  
Letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032092.png" />,
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Assume now that $a _ { 2 } > 1$ and let $n \geq m \geq 2$. For all $i \in \mathbf{N}$ there exists a $k  =  k  ( i ) \in \mathbf{N}$ such that $k \operatorname { log } m \leq i \operatorname { log } n < ( k + 1 ) \operatorname { log } m$. Since $( a _ { j } ) _ { j = 1 } ^ { \infty } $ is an increasing sequence, one concludes from $a _ { n^i }  = ( a _ { n } )^i$ that $( a _ { m } ) ^ { k } \leq ( a _ { n } ) ^ { i } \leq ( a _ { m } ) ^ { k + 1 }$ or, equivalently, that $k \operatorname { log } a _ { m } \leq i \operatorname { log } a _ { n } \leq ( k + 1 ) \operatorname { log } a _ { m }$. This yields
  
<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/b120320/b12032093.png" /></td> </tr></table>
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\begin{equation*} \frac { k } { k + 1 } \frac { \operatorname { log } a _ { \mathfrak { m } } } { \operatorname { log } m } \leq \frac { \operatorname { log } a _ { \mathfrak { n } } } { \operatorname { log } n } \leq \frac { k + 1 } { k } \frac { \operatorname { log } a _ { \mathfrak { m } } } { \operatorname { log } m }. \end{equation*}
  
It is clear that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032094.png" /> does not depend on the special choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032095.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032096.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032097.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032098.png" />, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b12032099.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320100.png" />. Consequently,
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Letting $ k  \rightarrow \infty$,
  
<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/b120320/b120320101.png" /></td> </tr></table>
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\begin{equation*} \frac { 1 } { p } : = \frac { \operatorname { log } a _ {  m  }} { \operatorname { log } m } = \frac { \operatorname { log } a _ { n } } { \operatorname { log } n }\; \text { for all } m , n \geq 2. \end{equation*}
  
From <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320102.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320103.png" /> it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320104.png" />. This completes the proof.
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It is clear that $p$ does not depend on the special choice of $m \in \mathbf{N}$. Moreover, $a _ { m } = m ^ { 1 / p }$ for all $m \in \mathbf{N}$. Since $a _ { n  + m} = F ( a _ { n } , a _ { m } )$, it follows that $F ( m ^ { 1 / p } , n ^ { 1 / p } ) = ( n + m ) ^ { 1 / p }$ for all $n \in \mathbf N$. Consequently,
  
Bohnenblust's theorem has some interesting consequences. For instance, T. Ando [[#References|[a1]]] used it to prove that a Banach lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320105.png" /> is isometrically isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320106.png" /> for some measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320107.png" />, or to some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320108.png" />, if and only if every closed sublattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320109.png" /> is the range of a positive contractive projection.
+
\begin{equation*} F ( s , t ) = ( s ^ { p } + t ^ { p } ) ^ { 1 / p }. \end{equation*}
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 +
From $F ( t , 1 - t ) = \| t x + ( 1 - t ) y \| \leq 1$ for all $0 \leq t \leq 1$ it follows that $p \geq 1$. This completes the proof.
 +
 
 +
Bohnenblust's theorem has some interesting consequences. For instance, T. Ando [[#References|[a1]]] used it to prove that a Banach lattice $E$ is isometrically isomorphic to $L ^ { p } ( \mu )$ for some measure space $( \Omega , \mathcal A , \mu )$, or to some $c_0 ( \Gamma )$, if and only if every closed sublattice of $E$ is the range of a positive contractive projection.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Ando,  "Banachverbände und positive Projektionen"  ''Math. Z.'' , '''109'''  (1969)  pp. 121–130</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.F. Bohnenblust,  "An axiomatic characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320110.png" />-spaces"  ''Duke Math. J.'' , '''6'''  (1940)  pp. 627–640</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Kakutani,  "Concrete representation of abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120320/b120320111.png" />-spaces and the mean ergodic theorem"  ''Ann. of Math.'' , '''42'''  (1941)  pp. 523–537</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Meyer-Nieberg,  "Banach lattices" , Springer  (1991)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Zippin,  "On perfectly homogeneous bases in Banach spaces"  ''Israel J. Math.'' , '''4 A'''  (1966)  pp. 265–272</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  T. Ando,  "Banachverbände und positive Projektionen"  ''Math. Z.'' , '''109'''  (1969)  pp. 121–130</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  H.F. Bohnenblust,  "An axiomatic characterization of $L _ { p }$-spaces"  ''Duke Math. J.'' , '''6'''  (1940)  pp. 627–640</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  S. Kakutani,  "Concrete representation of abstract $L _ { p }$-spaces and the mean ergodic theorem"  ''Ann. of Math.'' , '''42'''  (1941)  pp. 523–537</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  P. Meyer-Nieberg,  "Banach lattices" , Springer  (1991)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M. Zippin,  "On perfectly homogeneous bases in Banach spaces"  ''Israel J. Math.'' , '''4 A'''  (1966)  pp. 265–272</td></tr></table>

Latest revision as of 18:49, 26 January 2024

Consider the space $L ^ { p } ( \mu )$, $1 \leq p \leq \infty$, and a measure space $( \Omega , \mathcal A , \mu )$. Since the norm $\| \cdot \| p$ is $p$-additive, it is easily seen that the following condition is satisfied: For all $x , y , u , v \in L ^ { P } ( \mu )$ satisfying $\| x \| _ { p } = \| u \| _ { p }$, $\| y \| _ { p } = \| v \| _ { p }$, $x \perp y$ and $u \perp v$ (in the sense of disjoint support), one has $\| x + y \| _ { p } = \| u + v \| _ { p }$.

In [a2] H.F. Bohnenblust showed that the spaces $L ^ { p } ( \mu )$ are the only Banach lattices (cf. also Banach lattice) possessing this property; more precisely, he proved the following theorem, now known as the Bohnenblust theorem: Let $E$ be a Banach lattice of dimension $\geq 3$ satisfying $\| x + y \| = \| u + v \|$ for all $x , y , u , v \in E$ such that $\| x \| = \| u \|$, $\| y \| = \| v \|$, $x \perp y$, and $u \perp v$. Then there exists a $p$, $1 \leq p \leq \infty$, such that the norm on $E$ is $p$-additive.

Here, for $p < \infty$, a norm is said to be $p$-additive if $\| x \| ^ { p } + \| y \| ^ { p } = \| x + y \| ^ { p }$ for all $x , y \in E$ with $x \perp y$; a norm is said to be $\infty$-additive, or, equivalently, $E$ is said to be an $M$-space, if $\| x + y \| = \operatorname { max } \{ \| x \| , \| y \| \}$ for all $x , y \in E$ with $x \perp y$.

It should be noted that when $1 \leq p < \infty$, every Banach lattice with a $p$-additive norm is isometrically isomorphic to $L ^ { p } ( \mu )$, with $( \Omega , \mathcal A , \mu )$ a suitable measure space. This representation theorem is essentially due to S. Kakutani [a3], who considered the case $p = 1$; the proof of the more general result follows almost the same lines. For $p = \infty$ the situation is not so clear: there exist many $M$-spaces that are not isomorphic to any concrete $L ^ { \infty } ( \mu )$-space, for instance $c_0$.

In the proof of his theorem, H.F. Bohnenblust introduced an interesting and tricky method to construct the $p$ such that the norm is $p$-additive. A similar method was used later by M. Zippin [a5] to characterize $\mathbf{l}^{p}$-spaces in terms of bases. Since the proof given by Bohnenblust is interesting in itself, the main ideas are sketched below.

By hypothesis, there exists a function $F : [ 0 , \infty ) ^ { 2 } \rightarrow [ 0 , \infty )$ defined by

\begin{equation*} F ( s , t ) = \| t x + s y \| \text { for all } s , t \geq 0, \end{equation*}

whenever $x$ and $y$ are disjoint vectors of norm one. It can easily be verified that the function $F$ has the following properties:

1) $F ( 0 , t ) = t$;

ii) $F ( s , t ) = F ( t , s )$;

iii) $F ( r s , r t ) = r F ( s , t )$;

iv) $F ( r , F ( s , t ) ) = F ( F ( r , s ) , t )$;

v) $F ( s , t ) \leq F ( s _ { 1 } , t _ { 1 } )$ if $s\leq s_ 1$, $t \leq t_1$. The only non-trivial inclusion iv) follows from

\begin{equation*} F ( r , F ( s , t ) ) = \| r x + \| s y + t z \| z \| = \end{equation*}

\begin{equation*} = F ( s , t ) \left\| \frac { r } { F ( s , t ) } x + z \right\| = \end{equation*}

\begin{equation*} = F ( s , t ) \left\| \frac { r } { F ( s , t ) } x + \frac { 1 } { F ( s , t ) } ( s y + t z ) \right\| = \end{equation*}

\begin{equation*} = \| r x + s y + t z \| = F ( F ( r , s ) , t ) \end{equation*}

for all disjoint $x , y , z \in E _ { + }$ of norm one and $r , s , t \geq 0$ with $t \neq 0$.

One defines $a _ { 1 } = 1$ and $a _ { n + 1} = F ( 1 , a _ { n } )$ for all $n \geq 1$. Property v) implies that the sequence $( a _ { n } ) _ { n = 1 } ^ { \infty }$ is increasing. By induction one obtains $a _ { n + m} = F ( a _ { n } , a _ { m } )$ and . If $a _ { 2 } = 1$, then properties i)–v) easily imply $F ( s , t ) = \operatorname { max } \{ s , t \}$ for all $s , t \geq 0$. Hence, $E$ is an $M$-space.

Assume now that $a _ { 2 } > 1$ and let $n \geq m \geq 2$. For all $i \in \mathbf{N}$ there exists a $k = k ( i ) \in \mathbf{N}$ such that $k \operatorname { log } m \leq i \operatorname { log } n < ( k + 1 ) \operatorname { log } m$. Since $( a _ { j } ) _ { j = 1 } ^ { \infty } $ is an increasing sequence, one concludes from $a _ { n^i } = ( a _ { n } )^i$ that $( a _ { m } ) ^ { k } \leq ( a _ { n } ) ^ { i } \leq ( a _ { m } ) ^ { k + 1 }$ or, equivalently, that $k \operatorname { log } a _ { m } \leq i \operatorname { log } a _ { n } \leq ( k + 1 ) \operatorname { log } a _ { m }$. This yields

\begin{equation*} \frac { k } { k + 1 } \frac { \operatorname { log } a _ { \mathfrak { m } } } { \operatorname { log } m } \leq \frac { \operatorname { log } a _ { \mathfrak { n } } } { \operatorname { log } n } \leq \frac { k + 1 } { k } \frac { \operatorname { log } a _ { \mathfrak { m } } } { \operatorname { log } m }. \end{equation*}

Letting $ k \rightarrow \infty$,

\begin{equation*} \frac { 1 } { p } : = \frac { \operatorname { log } a _ { m }} { \operatorname { log } m } = \frac { \operatorname { log } a _ { n } } { \operatorname { log } n }\; \text { for all } m , n \geq 2. \end{equation*}

It is clear that $p$ does not depend on the special choice of $m \in \mathbf{N}$. Moreover, $a _ { m } = m ^ { 1 / p }$ for all $m \in \mathbf{N}$. Since $a _ { n + m} = F ( a _ { n } , a _ { m } )$, it follows that $F ( m ^ { 1 / p } , n ^ { 1 / p } ) = ( n + m ) ^ { 1 / p }$ for all $n \in \mathbf N$. Consequently,

\begin{equation*} F ( s , t ) = ( s ^ { p } + t ^ { p } ) ^ { 1 / p }. \end{equation*}

From $F ( t , 1 - t ) = \| t x + ( 1 - t ) y \| \leq 1$ for all $0 \leq t \leq 1$ it follows that $p \geq 1$. This completes the proof.

Bohnenblust's theorem has some interesting consequences. For instance, T. Ando [a1] used it to prove that a Banach lattice $E$ is isometrically isomorphic to $L ^ { p } ( \mu )$ for some measure space $( \Omega , \mathcal A , \mu )$, or to some $c_0 ( \Gamma )$, if and only if every closed sublattice of $E$ is the range of a positive contractive projection.

References

[a1] T. Ando, "Banachverbände und positive Projektionen" Math. Z. , 109 (1969) pp. 121–130
[a2] H.F. Bohnenblust, "An axiomatic characterization of $L _ { p }$-spaces" Duke Math. J. , 6 (1940) pp. 627–640
[a3] S. Kakutani, "Concrete representation of abstract $L _ { p }$-spaces and the mean ergodic theorem" Ann. of Math. , 42 (1941) pp. 523–537
[a4] P. Meyer-Nieberg, "Banach lattices" , Springer (1991)
[a5] M. Zippin, "On perfectly homogeneous bases in Banach spaces" Israel J. Math. , 4 A (1966) pp. 265–272
How to Cite This Entry:
Bohnenblust theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohnenblust_theorem&oldid=16687
This article was adapted from an original article by Peter Meyer-Nieberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article