Let $X$ be a Banach space and let $Y$ be the kernel of a bounded linear functional on $X$, say $M: X \rightarrow \mathbb{R}$. Is it possible to upper-bound the dimension of the quotient space $X/Y$. I am new to analysis therefore really not sure on whether this is even a good question to consider. But I am working on a problem where an upper bound on the dimension of $X/Y$ could help.
Any help or insights would be greatly appreciated.