Bohnenblust theorem
Consider the space , 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 |
Bohnenblust theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohnenblust_theorem&oldid=55332