Quadratic Minimization

Question

Consider a functional $I\colon H \to R$ on $H$ Banach space, sufficiently regular. Is in generally true that $$ \inf_{\rho \in H}{I^2(\rho)}=\Big(\inf_{\rho \in H}{I(\rho)} \Big)^2 \quad ? $$ If not, when the equality is satisfied? In my opinion is easy to see that, assuming that exist minimum, the equality hold (usin $\min$) but without this hyphotesis i don't know how to work: maybe the Jensen inequality.. but on the other hand?

Answer 1

As Batman pointed out, you need to assume $I$ is nonnegative. Under this assumption, the equality follows from a general fact about real numbers:

If $f:[0,\infty)\to[0,\infty)$ is an increasing continuous function, then for every nonempty set $A\subset [0,\infty)$ we have $\inf f(A) = f(\inf A)$.

Answer 2

No. take $H = \mathbb{R}$ and $I: \mathbb{R} \to \mathbb{R}$ by $I(x) = x$. Then, the LHS is zero. the RHS is infinite.

To show equality, you'd have to have $I$ be non-negative.