Theorem (Erdős–Kaplanski) Let $F$ be a field, $V$ an infinite-dimensional vector space over $F$. Then $\dim V^*=|V^*|$.
I have read the proof here by Pierre-Yves Gaillard, but I can't understand the notation. In particular, what does $$K_1:=K_0(\{b_j\ |\ b\in B,\ j\in\mathbb N\})$$ mean and why $|K_1|<|K|$? I guess $b_j$ means the $j$th coordinate of $b\in K^\mathbb{N}$, and $K_0(X)$ means the subfield generated by $K_0\cup X$. However, I can't see how $|K_1|<|K|$. Can someone explain? I will really appreciate your help.
I agree with your interpretation that $b_j$ is the $j$-th coordinate of $b\in B$, as such interpretation seems to work in that proof. I think $|K_1|<|K|$ is apparent, since the prime field $K_0$ has cardinality no larger than $\aleph_0$ and thus $$|K_1|\leqslant|K_0(x_1,\cdots,x_n,\cdots)|=\aleph_0=|B|<|K|,$$ where $x_1,\cdots,x_n\cdots$ are indeterminates over $K_0$.