For those not familiar with the Energy Casimir method, I will quickly summarize the important steps.
The goal is to construct a norm for which nonlinear stability of a dynamical system $\dot u= X(u)$, where X is some nonlinear operator, can be achieved.
For this, one starts with a constant of motion called $H_C$ which has a critical point at the equilibrium $u_e$.
Then, one has to find a quadratic form $Q$ such that the following convexity estimates hold: \begin{gather} Q(\Delta u)\leq H_C(u_e+\Delta u)-H_C(u_e)-DH_C(u_e)\cdot \Delta u\\ Q(\Delta u)>0 \end{gather} for all $\Delta u\neq 0$.
The norm in which nonlinear stability can be achived is defined by \begin{equation} \|v\|^2:=Q(v) \end{equation} Now I wondered why this actually defines a norm. Positivity of the norm is guaranteed by the second convexity estimate and $\|0\|$ is indeed zero because of the first convexits estimate. Absolute scalability follows since the square of the norm was to be defined a quadratic norm. However, I am not sure how to formally show the subbadditivity property in infinite dimensions. In the finite dimensional case of n dimensions one can simply use the ansatz $Q(u)=\sum^{i+j=2}_{1\leq i,j\leq n} a_{ij}u_i u_j$ and then use the hölder inequality applied to a counting measure, but that doesnt work in infinite dimensions?
Any advice is appreciated