Let $(X,\tau_+)$ be a Banach space, consider $\tau_-$ the weakest topology such that all linear $\tau_+$-continuous functional are continuous with respect to $\tau_-$, finally consider $\Sigma_+$ (respectively $\Sigma_-$) the $\sigma$-algebra generated by $\tau_+$ (respectively $\tau_-$). Within this setup I have two questions
Is $\Sigma_-$ the smallest $\sigma$-algebra such that all $\Sigma_+$-measurable linear functional are measurable ?
Are all $\Sigma_-$-measurable linear functional $\tau_-$-continuous ?
Some thoughts :
It is clear that the set of $\tau_-$-continuous linear functionals is the same as the set of $\tau_+$-continuous linear functionals. If the first point mentioned above is true, then the set of $\Sigma_-$-measurable linear functionals and the set of $\Sigma_+$-measurable linear functionals are also the same and therefore the second point is equivalent to
Are all $\Sigma_+$-measurable linear functional $\tau_+$-continuous ?
In other-words, is any measurable (w.r.t. the Borel $\sigma$-algebra) linear functional on a B-space continuous ?
I have this feeling that the second point is false unless we add boundedness :
Are all bounded $\Sigma_-$-measurable linear functional $\tau_-$-continuous ?
which would be also very interesting to study.