Stimulated by the classical transposition method in PDEs, we introduced a new notion of solution, i.e., transposition solution to backward stochastic evolution equations (BSEEs for short). This is something like the generalized function solutions to PDEs. We obtain the well-posedness of BSEEs in general filtration space, without using the Martingale Representation Theorem. Among others, as an application of our transposition method, we establish a Pontryagin-type maximum principle for optimal controls of general infinite dimensional nonlinear stochastic evolution equations, in which both drift and diffusion terms can contain the control variables, and the control domains are allowed to be nonconvex.