Flats of a linear space

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Let be a linear space. A subset is called a flat of if for all with

Of course, the intersection of flats is another flat, hence each subset is contained in a smallest flat <E>. In particular, the set of all flats, ordered by inclusion, forms a complete lattice.

The rank of a linear space is defined to be

If has finite rank, we define the dimension of as

Contributed by Hauke Klein
Version $Id: flats.html,v 1.1 2001/01/02 19:41:05 hauke Exp $