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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?

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The second answer is obviously false, for large enough $\lambda$.

For $\lambda>\frak c$ note that the dimension of $X$ must be $\lambda$, therefore it has a basis of size $\lambda$.

Let $B$ be such basis, take $B'=\left\{\frac1{\|v\|}v\mid v\in B\right\}$, then $B'$ is a subset of the unit sphere, and it is a basis of $X$ and so linearly independent.

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I'm afraid this set is linearly dependent. – Jorg Mar 17 '13 at 1:21
Oops. I read "must it be linearly independent". – Asaf Karagila Mar 17 '13 at 1:22
@usao: I edited my answer. – Asaf Karagila Mar 17 '13 at 1:24

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