Notations: V denotes a vector space A, B, C, D denote subspaces of V respectively ≈ denotes the isomorphic relationship of the left and right operand dim(?) denotes the dimension of "?" Question: Find a vector space V and decompositions: V = A ⊕ B = C ⊕ D with A≈C but B and D are not isomorphic. My opinion: dim(V)=dim(A)+dim(B)=dim(C)+dim(D) and dim(A)=dim(C), but dim(B)≠dim(D) since V may not be finite-dimensional. It's an idea not an example, would you make a concrete example of V? Thanks for any help!