James room

The James space , named after Robert C. James and introduced in 1951, is a special vector space considered in mathematics . It is a Banach space that is isometrically isomorphic to its dual space without being reflexive . For a long time this property has been thought impossible. The James room can also be used to construct other examples.

definition

As a set, the James space is contained in the sequence space of the real zero sequences . For a sequence, define as a measure of the variation in the terms of the sequence ${\displaystyle J}$ ${\displaystyle c_{0}}$ ${\displaystyle (\alpha _{n})_{n}\in c_{0}}$ ${\displaystyle \|(\alpha _{n})_{n}\|_{a}\in [0,\infty ]}$

${\displaystyle \|(\alpha _{n})_{n}\|_{a}\,:=\,{\frac {1}{\sqrt {2}}}\sup \left\{\left(\sum _{n=1}^{m-1}(\alpha _{p_{n}}-\alpha _{p_{n+1}})^{2}+(\alpha _{p_{m}}-\alpha _{p_{1}})^{2}\right)^{\frac {1}{2}};\,m\geq 2,\,p_{1}<\ldots <p_{m}\right\}.}$

The supremum is formed over all natural numbers and all strictly ascending sequences of natural numbers. Finally be ${\displaystyle m\geq 2}$ ${\displaystyle p_{1}<\ldots <p_{m}}$

${\displaystyle J:=\{(\alpha _{n})_{n}\in c_{0};\,\|(\alpha _{n})_{n}\|_{a}<\infty \}.}$

${\displaystyle J}$is thus the set of real zero sequences whose fluctuation is limited in the sense of the number . For example, the episode is not in . ${\displaystyle (\alpha _{n})_{n}}$ ${\displaystyle \|(\alpha _{n})_{n}\|_{a}}$ ${\displaystyle (\alpha _{n})_{n}=(1,-1,{\frac {1}{\sqrt {2}}},-{\frac {1}{\sqrt {2}}},{\frac {1}{\sqrt {3}}},-{\frac {1}{\sqrt {3}}},\ldots ,{\frac {1}{\sqrt {n}}},-{\frac {1}{\sqrt {n}}},\ldots )}$ ${\displaystyle J}$

It can be shown that a vector space. Componentwise operations is related and that a standard is that a Banach space makes. This is the so-called James room. ${\displaystyle J}$ ${\displaystyle \|\cdot \|_{a}}$ ${\displaystyle J}$

Base in J

Let the -th unit vector in , that is , with the 1 in the -th position. It can be shown that a monotone shrinking basis in and therefore applies. ${\displaystyle e_{n}}$ ${\displaystyle n}$ ${\displaystyle J}$ ${\displaystyle e_{n}=(0,\ldots ,0,1,0,\ldots )}$ ${\displaystyle n}$ ${\displaystyle (e_{n})_{n}}$ ${\displaystyle J}$ ${\displaystyle \|(\alpha _{n})_{n}\|_{a}=\lim _{m\to \infty }\|\sum _{n=1}^{m}\alpha _{n}e_{n}\|_{a}=\sup _{m\in \mathbb {N} }\|\sum _{n=1}^{m}\alpha _{n}e_{n}\|_{a}}$

Double room

Proceeding from the properties of the basis , one can show that the canonical embedding in the bidual space is not surjective, more precisely the codimension of in is equal to 1, that is . ${\displaystyle (e_{n})_{n}}$ ${\displaystyle Q:J\rightarrow J^{''}}$ ${\displaystyle Q(J)}$ ${\displaystyle \,J^{''}}$ ${\displaystyle J^{''}/Q(J)\cong \mathbb {R} }$

${\displaystyle J}$is therefore not reflexive. Nevertheless, it is possible to construct an isometric isomorphism between and . The evidence is very technical and so will not be discussed further here. ${\displaystyle \,J}$ ${\displaystyle \,J^{''}}$

Counterexamples

The James room can be used to construct a number of counterexamples. The above observation shows that a Banach space that is isometrically isomorphic to its dual space is not necessarily reflexive, which refutes an older conjecture.

Many infinite-dimensional Banach spaces have this property . All infinite-dimensional Hilbert spaces have this property, because according to Fischer-Riesz's theorem , these are isomorphic to for infinite , and it is . For the sequence space , too , it is easy to see that there is an isometric isomorphism . ${\displaystyle X}$ ${\displaystyle X\oplus X\cong X}$ ${\displaystyle \ell ^{2}(I)}$ ${\displaystyle I}$ ${\displaystyle \ell ^{2}(I)\oplus \ell ^{2}(I)\cong \ell ^{2}(I\times \{0,1\})\cong \ell ^{2}(I)}$ ${\displaystyle c_{0}}$ ${\displaystyle ((\alpha _{n})_{n},(\beta _{n})_{n})\mapsto (\alpha _{1},\beta _{1},\alpha _{2},\beta _{2},\ldots )}$ ${\displaystyle c_{0}\oplus c_{0}\cong c_{0}}$

This does not apply to the James room, because one can show that in the case should also apply, which is obviously not the case. ${\displaystyle J\oplus J\cong J}$ ${\displaystyle \mathbb {R} \cong J^{''}/Q(J)\cong (J\oplus J)^{''}/Q(J\oplus J)\cong J^{''}/Q(J)\oplus J^{''}/Q(J)\cong \mathbb {R} ^{2}}$

A -Banach space can be made into a real vector space by restricting the scalar multiplication . is an example of a real Banach space that is not isomorphic to one for a complex Banach space . If it were , it could not be reflexive either, would have at least the complex codimension 1 and therefore the real codimension 2 in , but the real codimension of in bidual is 1. ${\displaystyle \mathbb {C} }$ ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle X_{\mathbb {R} }}$ ${\displaystyle J}$ ${\displaystyle X_{\mathbb {R} }}$ ${\displaystyle X}$ ${\displaystyle J\cong X_{\mathbb {R} }}$ ${\displaystyle X}$ ${\displaystyle Q(X)}$ ${\displaystyle \,X^{''}}$ ${\displaystyle Q(J)}$

The James room is also an example of a Banach room with a shudder base that does not have an unconditional base. That J has no unconditional basis follows from the fact that the bidual space of an infinite-dimensional Banach space with unconditional basis is not separable , but is separable because it is and is 1-codimensional in . ${\displaystyle J^{''}}$ ${\displaystyle J}$ ${\displaystyle J\cong Q(J)}$ ${\displaystyle J^{''}}$

Individual evidence

1. ↑ James A non-reflexive Banach space isometric with its second conjugate space , Proceedings National Academy of Sciences, Vol. 37, 1951, pp. 174-177, online, PDF file
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory , Springer New York (1998), ISBN 0-387-98431-3 , Chapter 4.5 - James Space