Banach spaces locally having a basis

Question

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.

Answer 1

The purpose of this post is to make an expanded summary of the comments to OP on request of Professor Gonzalez.

Local basis structure (LBS) was considered by Pujara in 1975 under a different name as pointed out by Prof. Gonzalez in his comment . To the best of my knowledge, both questions were answered negatively by Szarek in 1987. The snippet below is from [Szarek1987].

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.

Perhaps Szarek's construction should be read along with the asymptotically Hilbertian spaces in [CasazzaGarciaJohnson2001].

For the future visitors to this post: interested reader should pay attention to the type and cotype of the constructed spaces in both articles.