Limit of non-degenerate biliniear forms

Question

Let $X$ be a Banach space (in my specific case I have $X=C_b(\mathbb{R})$) and let $\{B_n\}_{n\geq 1}$ be a sequence of biliniear forms $B_n:X\times X\rightarrow \mathbb{C}$, which are non-degenerate in the first coordinate in the sense that $$ (\forall y\in X: B_n(x,y)=0)\Rightarrow x=0$$ holds for all $n\geq 1$. The bilinear form need not be symmetric, so I do not know if it is non-degenerate in the second coordinate. Moreover, I know that for $n<m$ \begin{equation}\begin{array}{lcl}|B_n(x,y)-B_m(x,y)|&\leq &\Big(\sum\limits_{j=n+1}^m exp(-j)\Big)||x||\cdot ||y||\\[0.2cm] &=&\frac{exp(-m)-exp(-n)}{1-e}||x||\cdot||y||\end{array}\end{equation} for any $x,y\in X$. Then $\lim\limits_{n\rightarrow\infty} B_n$ defines pointwise a bilinear form $B:X\times X\rightarrow \mathbb{C}$. My question is: is this new bilinear form $B$ also non-degenerate in the first coordinate in the above sense?

Answer 1

Giuseppe Negro's solution shows that this need not be the case.