  We present a unified framework for developing and analysing immersed finite
element (IFE) spaces for solving typical elliptic interface problems with
interface independent meshes. This framework allows us to construct a group of
new IFE spaces with either linear, or bilinear, or the rotated-Q1 polynomials.
Functions in these IFE spaces are locally piecewise polynomials defined
according to the sub-elements formed by the interface itself instead of its
line approximation. We show that the unisolvence for these IFE spaces follows
from the invertibility of the Sherman-Morrison matrix. A group of estimates and
identities are established for the interface geometry and shape functions that
are applicable to all of these IFE spaces. Most importantly, these fundamental
preparations enable us to develop a unified multipoint Taylor expansion
procedure for proving that these IFE spaces have the expected optimal
approximation capability according to the involved polynomials.
