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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?

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I can answer this question when the cardinality $\lambda$ of Hamel basis of $X$ is not less than $\mathfrak{c}$.

For any infinite dimensional Banach space its cardinality and cardinality of its Hamel basis are equal. See theorem 3.5 from The cardinality of Hamel bases of Banach spaces by Lorenz Halbeisen , Norbert Hungerbühler.

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 $X$.

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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.

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