An ascending chain of subspaces of a vector space. If is an dimensional vector space, a flag of is a filtration
(1)

where all inclusions are strict. Hence
(2)

so that . If equality holds, then for all , and the flag is called complete or full. In this case it is a composition series of .
A full flag can be constructed by fixing a basis of , and then taking for all .
A flag of any length can be obtained from a full flag by taking out some of the subspaces. Conversely, every flag can be completed to a full flag by inserting suitable subspaces. In general, this can be done in different ways. The following flag of
(3)

can be completed by switching in any line of the plane passing through the origin. Two different full flags are, for example,
(4)

and
(5)

Schubert varieties are projective varieties defined from flags.