Weak convergence notation in normed state spaces

Let $X$ be a normed space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Its bounded scalar-linear dual $X^{\ast }$ consists of the bounded $\mathbb{K}$-linear maps from $X$ to $\mathbb{K}$. For a sequence $(x_j)$ in $X$ and $x\in X$, write $x_j\rightharpoonup x$ in $X$, and say that $x_j$ converges weakly to $x$, if $\mathrm{\ell }(x_j)\to \mathrm{\ell }(x)$ in $\mathbb{K}$ for every $\mathrm{\ell }\in X^{\ast }$. A weakly null sequence is one that converges weakly to $0$.

For a complex normed space, its realification $X_{\mathbb{R}}$ is the same additive group and norm with scalar multiplication restricted to real scalars. Weak convergence in $X_{\mathbb{R}}$ uses all bounded real-linear maps $X_{\mathbb{R}}\to \mathbb{R}$ in the preceding definition.

The real and complex normed-space conventions are those of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms and Complex norm and complex normed space, respectively.

The dual and weak topology are those of The dual space X^* of a normed space and its dual norm and Weak topology on a normed space. By Weak convergence of nets and sequences, the displayed scalar-evaluation condition is exactly convergence of the sequence in the weak topology $\sigma (X,X^{\ast })$: each evaluation is continuous, and finitely many eventual scalar conditions suffice for a basic weak neighborhood. By Realification preserves a complex norm and its metric, the retained norm is a real norm on $X_{\mathbb{R}}$. The same argument applies to the real normed space $X_{\mathbb{R}}$. These conventions define convergence only; they assert no compactness or subsequence extraction.