Is $0$ the only vector in the kernel of every bounded linear functional?

Question

Let $X$ is a normed vector space, and let $x_0\in X$ have the property that for every bounded linear functional $f:X\rightarrow K$, $f(x_0)=0$. Then does $x_0=0$?

I think the answer is clearly yes, but I'm not sure how to prove it. How do you construct a bounded linear functional which is nonzero on a given nonzero vector?

Answer 1

Take a subspace $L=span\{x\}$ and then define $f:L\to K$ by $f(\lambda x)=\lambda$. This is clearly a bounded linear functional such that $f(x)\ne 0$. By Hahn-Banach theorem you can extend it to a bounded linear functional on $X$.

Answer 2

Use Hahn Banach to extend any non zero linear function defined on the vector space gnerated by $x_0$ to $X$.