Dual basis of Banach spaces?

Question

There is a notion of dual basis in the dual space of a Banach vector space of infinite dimension? It's possible to give an explicit definition of this dual basis as in the classical vector spaces (once fixed a basis in the space)? Maybe can you suggest me some references? Thank you!

Answer 1

First, you need to distinguish between Hamel basis, as defined in linear algebra, and schauder basis. And I don't know any book, which focuses on this aspect. An application of the Baire-theorem shows that any Hamel basis in a Banach space must be either finite or already uncountable.

Given a Hamel basis you can define 'dual elements'. However, these maps don't have to be continuous and don't constitute a basis for the (non-continuous) dual space. In fact, the dual space of an infinite-dimensional vector space (over the field $\mathbb{R}$ or $\mathbb{C}$) is always strictly larger than the original space.

If $H \subset X$ forms a Hamel basis for $X$, then the (non-continuous) dual space can be identified with all functions $f \colon H \rightarrow \mathbb{C}$. Its cardinality is $|\mathbb{K}|^{|H|} \ge 2^{|H|} > |H|$ by Cantor's theorem. (Note that we need here $|H| \ge |\mathbb{K}|$.)

The next example shows that the dual elements to a Schauder basis doesn't form a basis of the dual space: Take $X= l^1$, then the continuous dual space is $X^* = l^\infty$. As usal, $(e_n)_{n \in \mathbb{N}}$, where $e_n$ is the element with $1$ in the $n$-coordinate and otherwise zero, is a Schauder basis of $X$. Now we have the dual elements $e_n^*(x) = x_i$, which are continuous functionals, but don't form a basis for $l^\infty$. For example let $f(x) = \sum_{i=1}^\infty x_i$, i.e. $f = (1,1,1,1,\ldots)$, then $\|f- \sum_{k=1}^n h_k e_k^*|_\infty = 1$ for any finite sum $\sum_{k=1}^n h_k e_k^*$. That means that no linear combination of $(e_n^*)$ converges towards $f$.

All in all, you see that the concept of 'dual basis' doesn't make sense in the context of infinite dimensional banach spaces.