I have a simple question about the subadditivity of inf, it's easy to show that
$\inf_x (f+g)(x) \ge \inf_x f(x) + \inf_y g(y)$.
And for norm we have
$\inf_{x\in X,y\in Y}(\|x\|+\|y\|)\le \inf_{x\in X}\|x\| +\inf_{y\in Y}\|y\|$.
My qustion is why they look quite similar but with different directions of inequalities.
The second done is actually an equality. The first one would have been an equality if we had $\inf_{x,y} (f(x)+g(y))$ instead of $\inf_x (f+g)(x))$. The main difference is you have two variables $x$ and independent of each other in the second one whereas there is only one variable in the LHS of first inequality.