  An algebraic approach is developed to define and study infinite dimensional
grassmannians. Using this approach a quantum deformation is obtained for both
the ind-variety union of all finite dimensional grassmannians, and the Sato
grassmannian introduced by Sato. They are both quantized as homogeneous spaces,
that is together with a coaction of a quantum infinite dimensional group. At
the end, an infinite dimensional version of the first theorem of invariant
theory is discussed for both the infinite dimensional special linear group and
its quantization.
