In projective space $P^{n}$ two projective subspaces $P^{m_{1} }$ and $P^{m_{2} }$ are given. How are the dimensions of $P^{m_{1} }\cap P^{m_{2} }, P^{m_{1} }, P^{m_{2} }$ are connected with dimension of $Span(P^{m_{1} } \cup P^{m_{2} })$? The projective span Span (M) of a subset M of $P^{n}$ is the smallest projective subspace of $P^{n}$ containing M.
I suppose that the connection between are suchlike: m1 + m2 + 1. Am I right?
Well, if $V$ is a vector space, the (resp. non-empty) projective subspaces in $\Bbb PV$ are the ones of the form $\Bbb PW$ for a (resp. non-zero) linear subspace $W$ of $V$, with the obvious good behaviour with respect to generation and intersection. Moreover, $\dim \Bbb PW=\dim W-1$ (it also extends to $\dim \emptyset=\dim \{0\}-1=-1$).
By Grassmann, $$\dim (W_1+W_2)+\dim(W_1\cap W_2)=\dim W_1+\dim W_2$$and therefore $$\dim\langle \Bbb PW_1,\Bbb PW_2\rangle +\dim(\Bbb PW_1\cap \Bbb PW_2)+\not 2=\dim \Bbb PW_1+\dim \Bbb PW_2+\not 2$$ which is actually the same relation. For instance, two non-intersecting lines generate a subspace of dimension $3$, because $$\dim\langle r_1,r_2\rangle=\dim r_1+\dim r_2-\dim(r_1\cap r_2)=1+1-(-1)=3$$