Existence of particular functionals in a family of linear functionals

Question

Let $U\subset B$ be a subset of a Banach space $B$, and let $D$ be a complete topological vector space. I have given a family $\mathcal L(U)=\{L_u\ |\ u\in U\}$ of linear functionals $L_u:D\to\mathbb R$ and a function $c:D\to[0,\infty)$ such that \begin{align*} |L_u(x)-L_v(x)|\leq c(x)||u-v||\qquad\forall u,v\in B,x\in D. \end{align*} The question now is:

Is there any chance of deciding whether $\mathcal L(U)$ contains the zero functional? (i.e. is there some $u\in U$ such that $L_u(x)=0$ for each $x\in D$?)

In particular,

1. which (additional) conditions have to be satisfied? Or
2. how must $U$ or even $D$ look like?

Any hints or suggestions are highly appreciated. Thanks in advance!