I read that in any locally convex topological space $X$, not necessarily a Hausdorff space but with linear operations continuous, for any $x_0\ne 0$ we can define a continuous linear functional $f:X\to K$ such that $f(x_0)\ne 0$.
I cannot find a proof of that anywhere and cannot prove it myself. Please correct me if I am wrong, but I think that if $A$ is closed in $X$ and $x_0\in A$ there exist a continuous linear functional rigorously separating $x_0$ and $A$, but I am not sure whether we can use that... Thank you so much for any help!
In locally convex spaces, you can use the Hahn–Banach theorem. Take a functional on the one-dimensional subspace spanned by $x_0$ and extend to the whole space.
More generally, in a topological vector space $V$ you will find such a functional if and only if $V$ contains a convex, open proper subset which contains $x_0$.