Let $V \subset H$ be a continuous, compact and dense embedding with $V$ and $H$ Hilbert spaces.

Let $\beta_j:[0,T] \to \mathbb{R}$ be functions for each $j$, and let $v_j$ be a basis of $V_0$.

Suppose that for each $j$, we have $$\frac{d}{dt}h(t;\sum_{i=1}^\infty \beta_i(t)v_i, v_j) + a(t;\sum_{i=1}^\infty \beta_i(t)v_i, v_j) = 0$$ $$\beta_j(0) = \beta_j(T)$$ where $h(t;\cdot,\cdot):H \times H \to \mathbb{R}$ is an inner product on $H$ for each $t$, and $a(t;\cdot,\cdot):V \times V \to \mathbb{R}$ is an inner product on $V$ for each $t$.

I wish to show that this equation and this information implies that $\beta_j = 0$ (a.e) for all $j$.

I have tried a lot of things. The problem is I don't know how to tackle the first term in the equation, since the basis cannot be orthonormal wrt. $h(t;\cdot,\cdot)$ since $t$ is variable. So I cannot use the results of this thread (Uniqueness of solutions to an ODE system). And I can't multiply the equation by $\beta_i$ and sum up because I wouldn't know what to do with the first term.

Does anyone have any ideas?