7
votes
1answer
370 views

Non continuous Linear form on $E=C([0,1],\mathbb{R})$ without AC

Let's note $E=C([0,1],\mathbb{R})$ the Banach space of real continuous funtions from the [0,1] interval with the uniform norm. Is it possible to show a non-continuous linear form on $E$ exists ...
9
votes
1answer
302 views

Inequivalent complete norms and the axiom of choice

Hi, I've been wondering about the following : Is it possible, without the axiom of choice, to have two inequivalent complete norms on a vector space? All the examples of inequivalent complete norms ...
5
votes
0answers
1k views

Explicit element of $(\ell^{\infty})^* - \ell^1$? [duplicate]

Possible Duplicate: What’s an example of a space that needs the Hahn-Banach Theorem? It is well known that the dual of $\ell^{\infty}$ properly contains $\ell^1$ (over $\mathbb{N}$, ...