# Banach spaces locally having a basis

The $$\mathcal{L}_p$$-spaces ($$1\leq p \leq \infty$$) are Banach spaces $$X$$ such that there exists a constant $$\lambda$$ so that every finite dimensional subspace $$E$$ of $$X$$ is contained in another finite dimensional subspace $$F$$ which is $$\lambda$$-isomorphic to $$\ell_p^n$$.

Let us say that a Banach space $$X$$ locally admits a basis if there exists a constant $$\lambda$$ so that every finite dimensional subspace $$E$$ of $$X$$ is contained in another finite dimensional subspace $$F$$ admitting a basis with basis constant $$\leq \lambda$$.

Each $$\mathcal{L}_p$$-space locally admits a basis, and it is not difficult to show that if $$X$$ has a basis then it locally admits a basis.

Questions.

1. Suppose that $$X$$ is separable and locally admits a basis. Does $$X$$ have a basis?
2. Suppose that $$X$$ is separable. Does $$X$$ locally admit a basis?

The answer to Question 1 is positive for $$\mathcal{L}_p$$-spaces.

• Professor Gonzalez, as far as I can remember, both questions have a negative answer in Szarek's paper where he provided a space with BAP & without a basis. Mar 6 at 23:50
• Thank you for your comment. Mar 7 at 9:12
• I have looked at Szarek's paper. I think you should write an answer, trying to say something about the second question using Kalton's review in MathScinet to one of Pujara's papers. Mar 7 at 9:13
• I'm afraid I don't have an access to MathSciNet, and I don't have a copy of Kalton's review in another form. Presently I can record here the exact reference to Szarek's article doi.org/10.1007/BF02392555 Corollary 1.6 within nicely sums up the answers to both questions -- for future visitors to this MO post. Mar 9 at 1:20
• Szarek's result aside, perhaps Enflo's space is another example of a separable space with local basis structure (by Proposition 1.3 in doi.org/10.1007/BF02392555) & without basis? Mar 9 at 1:28

Imho, the main actor in Szarek's construction is the space $$(\bigoplus Y_{p_k}^{n_k})_{\ell^2}$$ that lacks LBS, where $$Y_p^n$$ are carefully chosen $$n$$-dimensional subspaces of $$L^p$$ (Proposition 3.1) and $$p_k\to 2$$, $$n_k\to\infty$$ are carefully chosen sequences (Proposition 4.1). It is worth to note Remark 5.1 that points out other nonreflexive examples, cf. Professor Bill Johnson's comment.