In some stuff I'm reading on numerical methods, it keeps on talking about but never defining what it means for
a sequences of values $\{x^k\}$ to weakly converge to a fixed point of an operator
Can someone provide a reference as to the meaning of this term?
A topological vector space $X$ has a continuous dual space $X^*$; the space of all continuous linear functionals from $X$ to $\mathbb F$ where typically $\mathbb F = \mathbb R$ or $\mathbb F = \mathbb C$ ($\mathbb F$ is the field that $X$ is defined over). Ordinary convergence is convergence in the topology on $X$. Weak convergence is convergence along functionals in $X^*$. That is, a sequence $\{x_k\}$ in $X$ is said to converge weakly to $x \in X$ iff for every $f\in X^*$ we have $f(x_k) \to f(x)$ in $\mathbb F$.
If a sequence converges, then it also converges weakly to the same limit; the converse is not true. For example $\{\cos(2\pi i k x)\}$ converges weakly to zero in $L^2([0,1])$ but it does not have a classical limit.