Total subspaces with long chains of nowhere norming weak∗^* sequential closures

If a separable Banach space $X$ is such that for some nonquasireflexive Banach space $Y$ there exists a surjective strictly singular operator $T:X\to Y$ then for every countable ordinal $\alpha$ the dual of $X$ contains a subspace whose weak$^*$ sequential closures of orders less than $\alpha$ are not norming over any infinite-dimensional subspace of $X$ and whose weak$^*$ sequential closure of order $\alpha +1$ coincides with $X^*

Different forms of metric characterizations of classes of Banach spaces

For each sequence X of finite-dimensional Banach spaces there exists a sequence H of finite connected nweighted graphs with maximum degree 3 such that the following conditions on a Banach space Y are equivalent: (1) Y admits uniformly isomorphic embeddings of elements of the sequence X. (2) Y admits uniformly bilipschitz embeddings of elements of the sequence H.Comment: Accepted for publication in Houston Journal of Mathematic