From Wikipedia, the weak topology on a topological vector space $X$ is the initial topology with respect to its continuous dual $X^*$. In other words, it is the coarsest topology (the topology with the fewest open sets) such that each element of $X^*$ is a continuous function.
a net $(x_λ)$ in $X$ converges in the weak topology to the element $x$ of $X$ if and only if $φ(x_λ)$ converges to $φ(x)$ in $\mathbb R$ or $C$ for all $φ$ in $X^*$.
If I am correct, a net $(x_λ)$ in $X$ is said to converge weakly to $x$, if $φ(x_λ)$ converges to $φ(x)$ in $\mathbb R$ or $C$ for all $φ$ in $X^*$. Weak convergence therefore is a pointwise convergence, and has the topology of pointwise convergence on $X$ viewed as a set of linear functionals on $X^*$.
Does the quote above equivalently mean that the weak topology on $X$ is the same as the topology of pointwise convergence for weak convergence?
Thanks and regards!
