Give an example of a vector space VVV and subspacecs U1,U2U_1,U_2U1,U2 of VVV such that U1×U2U_1\times U_2U1×U2 is isomorphic to U1+U2U_1+U_2U1+U2 but U1+U2U_1+U_2U1+U2 is not a direct sum.
Hint: VVV must be infinite-dimensional
TODO