This passage is from a proof. They construct a functional and want to show it's not continuous on $X$.
Enumerate a countable basis $\{x_{k}\}$. Assume they are on the unit ball in $X$. For each natural number $k$ and vector $x \in X$ define $\psi_k(x)$ to be the coefficient of $x_k$ with respect to the expansion of $x$ in the Hamel basis. Then each $\psi_k$ belongs to the set of linear functionals on $X$ and therefore the functional $\psi: X \rightarrow \mathbb{R}$ defined by: $$ \psi(x) = \sum_{k=1}^{\infty}k\psi_k(x) \quad \forall x\in X$$ is also a linear functional on $X$. However, this linear functional is not bounded since each $x_k$ is a unit vector for which $\psi_k(x)=k$
Is this a typo? (i.e. should this read "...for which $\psi(x_k) = k$"?).
This might be an example of not great writing rather than a strict typo. Maybe you are meant to read it as "each $x_k$ is a unit vector such that taking $x = x_k$ yields $\psi_k(x) = k$" and it will follow that $\psi$ is unbounded. However the reasoning is really that $\psi(x_k) = k$, as you say.