Let G be one of the ind-groups GL(∞), O(∞),
Sp(∞) and P⊂G be a splitting parabolic
ind-subgroup. The ind-variety G/P has been identified with
an ind-variety of generalized flags in the paper "Ind-varieties of generalized
flags as homogeneous spaces for classical ind-groups" (Int. Math. Res. Not.
2004, no. 55, 2935--2953) by I. Dimitrov and I. Penkov. In the present paper we
define a Schubert cell on G/P as a B-orbit on
G/P, where B is any Borel ind-subgroup of
G which intersects P in a maximal ind-torus. A
significant difference with the finite-dimensional case is that in general
B is not conjugate to an ind-subgroup of P, whence
G/P admits many non-conjugate Schubert decompositions. We
study the basic properties of the Schubert cells, proving in particular that
they are usual finite-dimensional cells or are isomorphic to affine ind-spaces.
We then define Schubert ind-varieties as closures of Schubert cells and study
the smoothness of Schubert ind-varieties. Our approach to Schubert
ind-varieties differs from an earlier approach by H. Salmasian in "Direct
limits of Schubert varieties and global sections of line bundles" (J. Algebra
320 (2008), 3187--3198).Comment: Keywords: Classical ind-group, Bruhat decomposition, Schubert
decomposition, generalized flag, homogeneous ind-variety. [26 pages