Spectral norm inequality Suppose I have two matrices A, B 
$$
A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{m \times n}
$$
Then on what conditions on A and B will the followig ineqiality hold:
$$
||A+B||_2 \geq ||A||_2
$$
I for some reason feel that this would hold when the spaces spanned by the columns these matrices are different. But maybe I am wrong, or I cannot mathematically articulate it.   

You are wrong.  In particular, if we take
$$
A = \pmatrix{1&100\\0&0}\\
B = \pmatrix{-1&-100\\1/100&1}
$$
Verify that the column space of the two matrices is different, but $\|A + B\| < \|A\|$.
