Pitt theorem

The set of Pitt , named after Harry Raymond Pitt , is a set of the mathematical branch of functional analysis . It makes a statement about operators between the sequence spaces , from which it follows that the -spaces are pairwise non-isomorphic. ${\displaystyle \ell ^{p}}$ ${\displaystyle \ell ^{p}}$

Formulation of the sentence

• Be it . If a closed subspace is, every continuous , linear operator is compact . ${\displaystyle 1\leq p<r<\infty }$ ${\displaystyle X\subset \ell ^{r}}$ ${\displaystyle X\rightarrow \ell ^{p}}$

A weaker formulation is obtained if one only considers the subspace : ${\displaystyle X=\ell ^{r}}$

• For every continuous linear operator is compact. ${\displaystyle 1\leq p<r<\infty }$ ${\displaystyle \ell ^{r}\rightarrow \ell ^{p}}$

Applications

The simple consequence is that there can be no isomorphism for . By going over to the inverse of the isomorphism if necessary, one can assume without restriction . According to Pitt's theorem, such an isomorphism would have to be compact, from which the compactness of the unit sphere followed and thus a contradiction to the infinite dimension of the spaces involved. The above weaker formulation is sufficient for this. ${\displaystyle \ell ^{r}\rightarrow \ell ^{p}}$ ${\displaystyle r\not =p\in [1,\infty )}$ ${\displaystyle r>p}$

But the first formulation of Pitt's theorem gives much more. None of the -spaces is isomorphic to a closed subspace one for . Two infinite-dimensional Banach spaces are called completely incomparable if each closed, infinite-dimensional subspace of one is not isomorphic to a closed subspace of the other Banach space. For are so and completely incomparable. ${\displaystyle \ell ^{p}}$ ${\displaystyle \ell ^{r}}$ ${\displaystyle r\not =p}$ ${\displaystyle r\not =p}$ ${\displaystyle \ell ^{r}}$ ${\displaystyle \ell ^{p}}$

Using the Chinchin inequality one can easily see that every space L ^p ([0,1]) contains an isomorphic, closed subspace. Since this is for , does not apply to the above, is to not isomorphic if . In contrast, according to Fischer-Riesz's theorem , one even has an isometric isomorphism . ${\displaystyle \ell ^{2}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle p\not =2}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle L^{p}([0,1])}$ ${\displaystyle p\not =2}$ ${\displaystyle p=2}$ ${\displaystyle \ell ^{2}\cong L^{2}([0,1])}$

Individual evidence

1. ^ HR Pitt: A note on bilinear forms , J. London Math. Soc. (1932), Volume 11, pp. 174-180
2. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory , Springer-Verlag (2006), ISBN 978-1-4419-2099-7 , Theorem 2.1.4
3. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Theorem 4.23
4. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory , Springer-Verlag (2006), ISBN 978-1-4419-2099-7 , Definition 2.1.7