Importance of a basic sequence in Banach Space Theory

Question

In Classical Banach Spaces I and II by Lindenstrauss and Tzafriri, their first definition in page $1$ is as follows:

A sequence $\{x_n\}_{n=1}^\infty$ in a Banach space $X$ is called a Schauder basis of $X$ if for every $x\in X$ there is a unique sequence of scalars $\{a_n\}_{n=1}^\infty$ so that $x = \sum_{n=1}^\infty a_nx_n.$ A sequence $\{x_n\}_{n=1}^\infty$ which is a Schauder basis of its closed linear span is called a basic sequence.

In Topics in Banach Space Theory $2$nd edition by Albiac and Kalton, they quoted the following in page $6:$

As the reader will quickly realize, basis sequences are of fundamental importance in the theory of Banach spaces and will be exploited throughout this volume.

After reading the two sentences, I have the following question.

Question: Why are basic sequences so important that we need to have a definition for it?

Answer 1

Basic sequences are a useful tool for identifying the isomorphic structure of Banach spaces. In particular, a basic sequence that is not a basis generates a closed subspace of the ambient Banach space $X$ so if we have a basic sequence that is isomorphic to (say) $c_0$ or $\ell_1$ then we know $X$ possesses such subspaces. Much work has been done on complemented subspaces of Banach spaces, so all of this body of knowledge can then be brought to bear.

In fact, finding a Banach space ${\cal T}$ that has no subspace isomorphic to $c_0$ or $\ell_1$ was hard enough that the Tsirelson space wasn't found until $1974$, followed later by the Gowers-Maurey space. Moreover, there was a proposed Polymath project in $2009$ that asked if every 'explicitly defined' Banach space necessarily contained either $c_0$ or $\ell_1$.

Since a basic sequence is always dense in its closed linear span the subspace of $X$ that it generates must be separable (and note that Enflo's example of a non-separable Banach space with basis was only found in $1973$), and separable Banach spaces are usually easier to work with. Combining this with the complemented subspace work mentioned above, we again see that we can determine a lot about a Banach space from knowing about its basic sequences.

If a Banach space $X$ possesses a basis we cannot, unfortunately, conclude that $X^*$ has a basis. Indeed, let $X=\ell_1$; then $X^*=\ell_\infty$ which is non-separable and hence possesses no basis. However, whenever we have co-ordinate functionals $\{x^*_n\}_{n \in {\mathbb N}}$ for which $x^*_n(x) = x_n \quad \forall n \in {\mathbb N} \ \forall x \in X$ we do know that $\{x^*_n\}_{n \in {\mathbb N}}$ is a basic sequence in $X^*$. Also, a result of Mazur tells us that ever non-trivial weakly-null sequence possesses a basic subsequence, so it is apparant that basic sequences are easier to find that actual bases.

So: a basic sequence can tell us a lot about a Banach space and is often easier to find than a basis, which makes them important in the theorist's toolkit.