Show that any vector space \( \displaystyle V \) over a field \( \displaystyle \mathbb{K} \) has a (Hamel) basis.

Furthermore, show that any two bases of \( \displaystyle V \) have the same cardinality.

Additionally, show that if \( \displaystyle X \) is an infinite dimensional Banach space, then every Hamel basis of \( \displaystyle X \) is uncountable.