The weak topology on locally convex spaces

Let $X$ be a non-zero real vector space endowed with a locally convex topology $τ$; its continuous dual, $X^*$, is the vector space of all the linear and continuous functions on $X$. Consider the topology on $X^*$ defined as the coarsest locally convex topology on $X^*$ which makes continuous all the mappings $$T_x:X^*\rightarrow\mathbb{R},\quad T_x(g)=g(x),\quad g\in X^*, x\in X.$$

Is this topology the same as the weak$^*$-topology on $X^*$? In other words, is this the same as the coarsest (not necessarily locally convex) topology on $X^*$ which makes continuous all the mappings $T_x$ as described above?

I have never seen the weak$^*$-topology defined as "the coarsest locally convex ..." because the two ideas you mention do give the same topology. Indeed, if you consider the coarsest topology that makes the point evaluations continuous, you have that $f_j\to 0$ if and only if $f_j(x)\to0$ for all $x\in X$. This topology is determined by the seminorms $$p_x(f)=|f(x)|,\ \ x\in X,$$ since $f_j(x)\to0$ for all $x$ if and only if $p_x(f_j)\to0$ for all $x$. So it is locally convex.