By Shiing-Shen Chern

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**Extra resources for Complex Manifolds**

**Example text**

Often it is ad dit ionally required that t he modulus Izl := sup{Re (eil/z) : 0 :::; e :::; 1r} exist s for every element z E E EB iE. In th e case of a K -space or an a rbit ra ry Ban ach latt ice t his requirement is a utomat ically satisfied, since a complex K-space is the complexificat ion of a real K-space. Speakin g about order properties of a complex vector lat tic e E EB i E, we mean its real par t E . The conce pts of sublattice, ideal, band , pro ject ion , etc . are naturally t ra nslated to t he case of a compl ex vector latti ce by appropriate com plexificati on.

Kantorovich. This notion appeared in Kantorovich's first fundamental article on this topic [153J where he wrote, "In this note, I define a new type of space that I call a semi ordered linear space. " Here L. V. Kantorovich stated an important methodological principle, the heuristic transfer principle for K -spaces. (3) The concept of universal completion for a K-space was introduced in another way by A. G. Pinsker (see [163]). He also proved existence of a universal completion unique to within isomorphism for an arbitrary K-space.

Every element x E E admits the representation JAde~ , 00 x = - 00 where the integral is understood to be the I -uniform limit of the integral sum s x ((3 ) := LTn(e~n+l - e~J , nEZ t n :::; r« :::; t n + l , as 8((3) := sUPnEZ (t n+ l - tn) -. 0, where (3 := (tn) nEZ is a partition of th e real line. e. by eleme nt s of the form L ~=l Akek , where Al , . , An E lR a nd el, ... ,e n E \1:(e) . In the case when the latter holds in a vector lat ti ce E we say th at E possesses t he weak Freudenthal propert y.

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