Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer 1

up vote 0 down vote accepted

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.

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.