# Hamel basis and Banach spaces

Suppose $X$ is a linear space and $X$ has a Hamel basis with uncountable number of elements. Does there exist a norm on $X$ such that $X$ is a Banach space with respect to this norm?

I can answer this question when the cardinality $\lambda$ of Hamel basis of $X$ is not less than $\mathfrak{c}$.
Consider Banach space $\ell_\infty(\Lambda)$. Clearly $\operatorname{Card}(\ell_\infty(\lambda))=\lambda\times \operatorname{Card}(\mathbb{C})=\lambda$, so cardinalities of Hamel bases of $\ell_\infty(\lambda)$ and $X$ coincide. Therefore, there is some linear isomorphism between $X$ and $\ell_\infty(\Lambda)$. This isomorphism induces complete norm on $X$.
The cardinality $\lambda$ of a Hamel basis of an infinite-dimensional Banach space always satisfies $\lambda^\omega =\lambda$. Choose your favourite uncountable cardinal without this property.