I proved that for subspaces $U_i$ of $V$ the inequality $\mathrm{dim}(U_1 + ...+U_m) \le \mathrm{dim}(U_1) + ... + \mathrm{dim}(U_m)$ holds.
I proved it as follows (would you please tell me if my proof is correct): For two subspaces $U,W$ we know that $\mathrm{dim}(U+W) = \mathrm{dim}U + \mathrm{dim}W - \mathrm{dim}(U \cap W)$ so that the claim is true for two subspaces. Then the claim follows because $U_1 + ... + U_{m-1}$ and $U_m$ are two subspaces hence $\mathrm{dim}(U_1 + ...+U_{m}) \le \mathrm{dim}(U_1 + ...+U_{m-1}) + \mathrm{dim}(U_m)$. Apply the argument again to get $ \mathrm{dim}(U_1 + ...+U_{m-1}) \le \mathrm{dim}(U_1 + ...+U_{m-2}) + \mathrm{dim}U_{m-1}$. Apply the argument $m$-times to obtain the desired inequality.