I know that a Hamel basis can be dense in a Banach space (it was probably posted somewhere on this forum). I would like to construct a certain counter-example and doing this, I encountered the following problem (which might sound to be ad hoc).
Let $X$ be a non-separable Banach space. Is it possible to find a Hamel basis for $X$ consisting of unit vectors which is dense in the unit sphere?
Or, maybe the other extreme can happen:
Let $\lambda$ be a cardinal as big as you wish and let $X$ be a Banach space of cardinality $\lambda$. Suppose $A$ is contained in the unit sphere of $X$ and $|A|=\lambda$. Must $A$ be linearly dependent?