Weak star convergence

Let $X$ be a real or complex normed space and $(f_i{)}_{i\in I}$ a net in $X^{\ast }$ indexed by a nonempty directed preorder (Directed preorders and nets). For a specified $f\in X^{\ast }$, write if $f_i$ converges to $f$ in the weak-star topology. By Convergence and cluster points of a net in a topological space and Basic weak star neighborhoods, this means equivalently

$\mathrm{\forall }x\in X\mathrm{\forall }\epsilon >0\mathrm{\exists }{i}_0\in I\mathrm{\forall }i\ge i_0:\mathrm{\mid }{f}_i(x)-f(x)\mathrm{\mid }<\epsilon \mathrm{.}$

Topological convergence implies each displayed eventual condition by taking a one-evaluation neighborhood. Conversely, for a finite-evaluation neighborhood choose the finitely many eventual indices and take a common upper bound in $I$; past it all inequalities hold. An empty coordinate list imposes no condition. This proves both directions without any choice axiom. For sequences take $I=\mathbb{N}$.

The asserted limit belongs to the bounded dual: this definition does not identify an arbitrary pointwise limit of bounded functionals with a member of $X^{\ast }$. The limit, when it exists, is unique, since equality of all evaluations is equality of functions. Constant nets converge to their constant value, also when $X=\{0\}$.