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.**

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

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

8more comments