Dual vector families in normed vector spaces

Question

Suppose $X$ is a normed vector spaces, then $\{x_\alpha\}\subset X,\{f_\alpha\}\subset X^*\, (\alpha\in A)$ are called dual vector families if $f_\alpha(x_\beta)=\delta_{\alpha\beta}$ where $\delta$ denotes the Kronecker symbol.

Now, if $\{x_\alpha\}$ is a linearly independent family in $X$, prove that there must exist $\{f_\alpha\}\subset X^*$ such that they are dual families.

I've proved a weaker version (stronger conditions) using a variant of Hahn Banach. Suppose that $\forall \alpha$ we have $x_\beta\notin\overline{\text{span}}\{x_\beta\}_{\beta\ne\alpha}$, then for each $\alpha$ we readily have some $f_\alpha$ that assumes $1$ at $x_\alpha$ and $0$ in the closure of the span of the remaining vectors.

I'm not sure whether it is possible to generalise the result above. For example, what if $\{x_\alpha\}$ is an uncountable Hamel basis of $\ell^2$? Or, is it necessary to switch to another way?

Answer 1

Your approach is in the right direction, but your question in its general form has a negative answer. A counterexample is the following:

If $X$ is an infinite dimensional Banach space and $\{x_a: a\in A\}$ is a Hamel basis of $X$, then the dual family $\{f_a: a\in A\}$ is just the collection of the coordinate functionals and it is known that at most finitely many of them are bounded. So this collection basically belongs in $X^\#$, but not in $X^*$.

Therefore, if you want a positive result you must add some additional conditions on $\{x_a: a\in A\}$. The one that you wrote, for example, is a clever one. It also shows that in order to construct elements of $X^*$ with some desired properties, we don't have a lot of tools other than the theorem of Hahn - Banach.