Weak convergence of nets and sequences

Let $X$ be a real or complex normed space. Let $I$ be a nonempty directed preorder and $(x_i{)}_{i\in I}$ a net in $X$ (Directed preorders and nets). For $x\in X$, write $x_i\rightharpoonup x$ and say the net converges weakly to $x$ when it converges to $x$ in $\sigma (X,X^{\ast })$ (Weak topology on a normed space, Convergence and cluster points of a net in a topological space). Equivalently,

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

Indeed, topological convergence implies each of these eventual conditions because the inverse disk is a neighborhood. Conversely, a neighborhood contains a finite intersection of inverse scalar neighborhoods containing $x$. Coordinate convergence gives an eventual index for each of the finitely many conditions; directedness gives a common upper bound for them, after which the whole intersection contains the net. For the empty intersection use any index of the nonempty $I$. This proves the equivalence without a choice axiom.

A weakly convergent sequence is this definition with $I=\mathbb{N}$ in its usual order. A constant net converges weakly to its value, including in the zero space. The specified limit must be a point of $X$; without a separation hypothesis uniqueness is not part of the definition. Weak closure continues to mean topological closure, not merely the set of limits of sequences.