The sum of two projective spaces

Question

Let $K$ be a field and let $V$ be a vector space on $K$. We consider the projective space $P(V)$ on $V$ (the set of the vector subespaces $A$ in $V$ such that $\dim A = 1$). Let $S , T$ be two vector subespaces in $V$. By definition $P(S)$ and $P(T)$ are two projective spaces in $P(V)$. I want to prove that $P(S) + P(T)$ is a new projective space in $P(V)$ (the sum of two projective spaces is a new projective space). Obviously, $S + T$ is a vector space in $V$, so $P(S + T)$ is a projective space in $P(V)$. Must I use this argument to prove that $P(S) + P(T)$ is a projective space in $P(V)$? Is the equality $P(S) + P(T) = P(S + T)$ a definition? Thank you very much.

Answer 1

The expression $P(T) + P(S)$ does not make sense since there is no addition in $P(V)$. More precisely, the operation "$+$" does not pass to quotient.

So the only thing which wake sense is to look at $P(T + S)$ which is in fact the smallest projective space which contains both $P(T)$ and $P(S)$. But I won't call it $P(T) + P(S)$ since really the notion of addition does not exist when you do projectivize.