I am reading a lecture note on topological vector space: https://people.math.gatech.edu/~heil/6338/summer08/section9d.pdf
In Exercise E.17 on page 345, the author gives an iff condition of Hausdorffness. But I think the condition can be relaxed a little bit, that is, for any [MATH]x \ne 0[/MATH], there is an [MATH]\alpha\in J[/MATH] s.t. [MATH]\rho_\alpha(x)\ne0[/MATH]. To put in another way, we could have a seminorm taking nonzero value at zero, as long as at other points the above condition holds. Not very self-confident at math, I need a confirmation if I am right (no need to prove, just tell me if I am correct). Thank you.
In Exercise E.17 on page 345, the author gives an iff condition of Hausdorffness. But I think the condition can be relaxed a little bit, that is, for any [MATH]x \ne 0[/MATH], there is an [MATH]\alpha\in J[/MATH] s.t. [MATH]\rho_\alpha(x)\ne0[/MATH]. To put in another way, we could have a seminorm taking nonzero value at zero, as long as at other points the above condition holds. Not very self-confident at math, I need a confirmation if I am right (no need to prove, just tell me if I am correct). Thank you.
