We know that the Topological Dual of a vector space is strictly included in the algebraic dual. It mean that there are linear functional that are not continuous. In my course we construct the topological dual as follow : Let $\mathcal B$ a Banach space and $\mathcal B^*$ it's dual. Let $\mathcal T$ the thicker topology on $\mathcal B$ s.t. linear functional are continuous. We call $(\mathcal B,\mathcal T)$ the topological dual of $\mathcal B$.
Q1) Isn't it the weak topology of $\mathcal B$ instead of the dual topological ?
Q2) For me the dual topological is $\{f\in \mathcal B^*\mid f\text{ is continuous wrt the strong topology of }\mathcal B\}$. Do you agree ?