Although it's simply stated, this is neither a homework problem or trivial (I think, but I'd be happy to be proven wrong :) ).


Let $A,B$ be matrices and $x$ be a vector. Is it true that 
$$ \|P\_{A+B} x\| \geq \|P\_A x\| - \|P\_B x\|, $$
where $P\_A$ is the projection onto the range space of $A$?
(or is it true if you square the norms?)


I'm having difficulty even figuring out how to attack this: every attempt I've made falters on the facts that the range space of $A + B$ is not simply related to those of $A$ and $B$ and that the projection is nonlinear. Random instances haven't yet provided counterexamples to the inequality.



Does this norm inequality hold for projections onto the range of a sum of matrices?
Let
$$A=\pmatrix{1&0\cr 0&0}, B=\pmatrix{0&0\cr 1&0}, x=(x\_1,x\_2).$$
Then $P\_Ax=(x\_1,0)$, $P\_Bx=(0,x\_2)$, $P\_{A+B}x=((x\_1+x\_2)/2,(x\_1+x\_2)/2)$.
Thus you are asking if
$$|(x\_1+x\_2)/\sqrt{2}|\ge |x\_1|-|x\_2|.$$
Clearly, this is false in general. 


