Dvoretzky-Rogers theorem

The Dvoretzky-Rogers theorem , after Aryeh Dvoretzky and Claude Ambrose Rogers , is a theorem from the mathematical branch of functional analysis , which deals with the convergence behavior of series in Banach spaces .

Dvoretzky-Rogers' lemma

We begin with a lemma about finite-dimensional normalized spaces , which ensures the existence of a basis for which there is an estimate against the Euclidean norm of the coefficients :

• Lemma von Dvoretzky-Rogers : In a -dimensional normalized space there are vectors with norm 1, so that the following inequality holds for and all coefficients : ${\displaystyle n}$ ${\displaystyle E}$ ${\displaystyle x_{1},\ldots ,x_{n}\in E}$ ${\displaystyle 1\leq m\leq n}$ ${\displaystyle t_{1},\ldots ,t_{m}\in \mathbb {R} }$

${\displaystyle \|\sum _{i=1}^{m}t_{i}x_{i}\|\leq (1+{\sqrt {\frac {m(m-1)}{n}}})\cdot {\sqrt {\sum _{i=1}^{m}t_{i}^{2}}}}$.

The quality of the estimate depends on the number of summands, is the same in the worst case and therefore depends on the dimensions. If one wants to be independent of the dimension, one has to restrict the number of summands, as is done in the following corollary, which is the essential part of the proof of the Dvoretzky-Rogers theorem: ${\displaystyle m}$ ${\displaystyle 1+{\sqrt {n-1}}}$

• Corollary : If , there are vectors with norm 1 in every -dimensional normalized space , so that the following inequality applies to all coefficients : ${\displaystyle n\geq m(m-1)}$ ${\displaystyle n}$ ${\displaystyle E}$ ${\displaystyle x_{1},\ldots ,x_{m}\in E}$ ${\displaystyle t_{1},\ldots ,t_{m}\in \mathbb {R} }$

${\displaystyle \|\sum _{i=1}^{m}t_{i}x_{i}\|\leq 2\cdot {\sqrt {\sum _{i=1}^{m}t_{i}^{2}}}}$.

Dvoretzky-Rogers theorem

• Let be an infinite-dimensional Banach space and a sequence of positive numbers with . Then there exists a sequence of vectors from with , so that the series necessarily converges . ${\displaystyle E}$ ${\displaystyle (a_{n})_{n}}$ ${\displaystyle \Sigma _{n}a_{n}^{2}<\infty }$ ${\displaystyle (x_{n})_{n}}$ ${\displaystyle E}$ ${\displaystyle \|x_{n}\|=a_{n}}$ ${\displaystyle \Sigma _{n}x_{n}}$

To prove this, one obtains a suitable sequence of finite-dimensional subspaces from which one selects the sought vectors with the help of the above corollaries to the Dvoretzky-Rogers lemma.

Applications

A characterization of finite-dimensional spaces

After the set of Dvoretzky-Rogers there is a sequence in any infinite-dimensional Banach space with , so that the series necessarily converges, because we know that is true . Since it is diverging (see harmonic series ), the series is not absolutely convergent . So every infinite-dimensional Banach space contains an unconditionally convergent series that does not absolutely converge. Since unconditional and absolutely convergent series in finite-dimensional spaces coincide according to Steinitz's rearrangement theorem , the following characterization of finite-dimensional spaces is obtained, which is sometimes also referred to as Dvoretzky-Rogers' theorem. ${\displaystyle (x_{n})_{n}}$ ${\displaystyle \|x_{n}\|=1/n}$ ${\displaystyle \Sigma _{n}x_{n}}$ ${\displaystyle \Sigma _{n}1/n^{2}<\infty }$ ${\displaystyle \Sigma _{n}\|x_{n}\|}$

• A Banach space is finite-dimensional if and only if every unconditionally convergent series also converges absolutely.

A sentence by Orlicz

After a set of Władysław Orlicz applies to any necessarily convergent series in L ^p [0,1] , that where . Therefore a series with in L ² [0,1] cannot necessarily be convergent. This shows that the condition in Dvoretzky-Rogers' theorem cannot be weakened, because in this case the condition is even necessary. Conversely, the Dvoretzky-Rogers theorem shows that the initially unnatural restriction to exponents in Orlicz's theorem above is inevitable, because: ${\displaystyle \Sigma _{n}x_{n}}$ ${\displaystyle 1\leq p<\infty }$ ${\displaystyle \Sigma _{n}\|x_{n}\|^{r}<\infty }$ ${\displaystyle r:=\max(2,p)}$ ${\displaystyle \Sigma _{n}x_{n}}$ ${\displaystyle \Sigma _{n}\|x_{n}\|^{2}=\infty }$ ${\displaystyle \geq 2}$

• If in an infinite-dimensional Banach space the unconditional convergence of a series always follows for a fixed one , then we have . ${\displaystyle \Sigma _{n}x_{n}}$ ${\displaystyle \Sigma _{n}\|x_{n}\|^{r}<\infty }$ ${\displaystyle r}$ ${\displaystyle r\geq 2}$

Namely, if in the sequence space , then according to the Dvoretzky-Rogers theorem there is an unconditionally convergent series with for all , and for this series we have by assumption . This shows, and that means . ${\displaystyle (a_{n})_{n}}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle \Sigma _{n}x_{n}}$ ${\displaystyle \|x_{n}\|=|a_{n}|}$ ${\displaystyle n}$ ${\displaystyle \Sigma _{n}|a_{n}|^{r}=\Sigma _{n}\|x_{n}\|^{r}<\infty }$ ${\displaystyle \ell ^{2}\subset \ell ^{r}}$ ${\displaystyle 2\leq r}$

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• A. Dvoretzky and CA Rogers: Absolute and unconditional convergence in normed linear spaces , Proc. Nat. Acad. Sci. USA 36: 192-197 (1950)
• MI Kadets, VM Kadets: Series in Banach Spaces . Operator Theory: Advances and Applications, Vol. 94, Birkhäuser (1997), ISBN 978-3764354015 .