Please construct a counterexample for the following: $A$ is normed space and $M$ is a dense subspace of $A$, if there is a functional $f$ such that $f(M) = 0$, then $f=0$.
Besides, if $A$ is a Banach space, does the same conclusion hold?
Please construct a counterexample for the following: $A$ is normed space and $M$ is a dense subspace of $A$, if there is a functional $f$ such that $f(M) = 0$, then $f=0$.
Besides, if $A$ is a Banach space, does the same conclusion hold?
Consider functional $ f:c_{00}(\mathbb{N})\to\mathbb{C}:x\mapsto\sum\limits_{n=1}^{\infty}n x(n) $ where $c_{00}(\mathbb{N})$ is the space of finitely supported sequences with $\sup$ norm. Its kernel is dense in $c_{00}(\mathbb{N})$, however $f\neq 0$.