If V is the linear space over the field R of real numbers, and $U_1, U_2, U_3$ are subspaces of this space. Is the inequality always true: $\dim_R (U_1 + U_2 + U_3)\leq \dim_RU_1+\dim_RU_2+\dim_RU_3$
I tried to base my answer on what I know to be true:If $U_1, U_2, U_3$ are finite dimensional subspaces of a vector space $V$ , then$$\dim_R(U_1 + U_2 + U_3) = \dim_R(U_1) + \dim_R(U_2) + \dim_R(U_3)\\− \dim_R(U_1 ∩ U_2) − \dim_R(U_1 ∩ U_3) − \dim_R(U_2 ∩ U_3)$$.So my thinking is that if we take away the intersection subtraction the right side of the equation will always be greater or equal to the left side. Therefore the inequality is always right. Am I correct? Is that how you're supposed to do this?