Linear dimension of Banach spaces

Question

Let $X$ be some vector space (over $\mathbb{C}$). Note that if $X$ is of finite dimension we can identify $X$ with $\mathbb{C}^n$ for some natural $n$ and endow it with a norm $\|x\|=|x_1|+\ldots+|x_n|$. Thus $X$ becomes a banach space. However, if $X$ has a countable dimension, we cannot endow it with a banach norm! Really, suppose $\{x_1,x_2,\ldots\}$ is a countable Hamel base in $X$. It is easy to see that $X_n=\mathrm{Span}(x_1,\ldots, x_n)$ is closed and does not contain any ball. But $X$ is a union of $X_n$ hence $X$ cannot be a full metric space under Baire theorem!
So, we see that the linear dimension of banach space cannot be countable. I'd like to ask are there any other cardinal numbers $\mathfrak{K}$ such that no banach space has the linear dimension $\mathfrak{K}$.

Answer 1

See the article

H. Elton Lacey,
The Hamel Dimension of any Infinite Dimensional Separable Banach Space is $c$,
Amer. Math. Mon. 80 (1973), 298,

where a simple proof is given of the fact that every separable Banach space has linear dimension equal to the continuum.