On Zippin's Embedding Theorem of Banach spaces into Banach spaces with bases

TL;DR

This paper provides a new proof of Zippin's Embedding Theorem, showing that separable reflexive Banach spaces can be embedded into spaces with specific basis properties, leading to improved embedding results.

Contribution

The paper introduces a novel proof of Zippin's Embedding Theorem, enhancing existing embedding results for Banach spaces with bases.

Findings

Every separable reflexive Banach space embeds into one with a shrinking and boundedly complete basis.

Every Banach space with a separable dual embeds into one with a shrinking basis.

The new proof yields improved versions of related embedding theorems.

Abstract

We present a new proof of Zippin's Embedding Theorem, that every separable reflexive Banach space embeds into one with shrinking and boundedly complete basis, and every Banach space with a separable dual embeds into one with a shrinking basis. This new proof leads to improved versions of other embedding results.