# Importance of a basic sequence in Banach Space Theory

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?

• Schauder basis are like usual basis in finite dimensional spaces. It is very useful to have such a thing, they are generators of the space and give you an explicit form to write any element of it. Since you are also interested in subspaces this reasoning extends to basic sequences. There are likely other reasons why it is useful to find out that a sequence you are looking at has the property of being a basic sequence. Aug 6, 2017 at 10:09

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.