Abstract: The concepts of frame transform and orthogonal projection in Hilbert K-submodulesare are introduced in this paper. The Hilbert base, the (strong) complement of frames and the inner product of straight sum in Hilbert K-modules are redefined hereby. The (strong) complements of frames in Hilbert K-submodules are mainly studied from the aspect of orthogonal projection by the relationship between the frame transform and the orthogonal projection. Finally, some important results coming from the researches are obtained and generalized.

Key words: Hilbert K-submodule, Hilbert base, orthogonal projection, (strong) complement, orthogonal complement