Suppose $H$ is a Hilbert space. Is
$$\inf_{h \in H}\sup_{g \in H}\frac{f(h,g)}{|h||g|} \geq C$$
the same as
$$f(h,h) \geq C|h|^2\quad \forall h \in H$$
? Here $f\colon H \times H \to \mathbb{R}$ is continuous and linear and $|h|$ means the norm of $h$. Or does the first imply the second?
The second inequality implies the first. This is because $$sup_{g\in H}\frac{f(h,g)}{|h||g|}\geq \frac{f(h,h)}{|h||h|}\geq C$$ and taking the infimum over all $h$ gives the first inequality. However, the first inequality doesn't imply the second. For example, take the function $f$ defined by the matrix $$ f((x,y),(a,b))=(x,y)\left(\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right) \left(\begin{array}{cc} a\\ b \end{array}\right)$$ Then $f((x,y),(x,y))=0$ for all vectors so that in the second inequality the only possibility for $C$ is 0. On the other hand, you have $$\frac{f( (x,y),(-y,x) )}{x^2+y^2}=1$$ so that in the first inequality you can choose $C=1$.