What this note offers
A Lipschitz-free space turns a pointed metric space into a Banach space and a base-point-preserving Lipschitz map into a bounded linear operator. This note develops the construction and follows one proof route to the theorem that ℱ(H) contains no linearly isomorphic copy of c0 for every real Hilbert space H, without a separability assumption.
The organizing idea is a family of scalar Lipschitz tests whose increments have a uniform summability bound. Linearization turns such a family into one operator with values in ℓ1, where Schur's property controls weak convergence. The note uses this principle first to approximate a weakly null sequence on one compact set and then to treat compact convex subsets of Hilbert spaces.
The compact Hilbert-space argument is developed through explicit square-root formulas, finite Hilbert averages, finite subdivisions and multiplicative damping. Its estimates use algebraic identities and finite sums, rather than derivatives or integrals. Nearest-point retractions and a consequence of the Bessaga–Pełczyński theorem complete the passage to the whole Hilbert space.
This is a detailed reconstruction of existing arguments for a specific conclusion, not a comprehensive survey or a priority claim. The stronger results in the cited literature, including weak sequential completeness and property (V*), are distinguished from what is proved here. The note can be read independently of the companion discussion of Lipschitz homeomorphic, linearly non-isomorphic Banach spaces.
Intended readers
The note is for graduate students and researchers comfortable with Banach space basics, bounded operators, weak convergence and elementary Hilbert space geometry. It assumes Hahn–Banach, Schur's property of ℓ1 and the Bessaga–Pełczyński theorem on unconditional series. Lipschitz-free space duality, linearization and the specific compactness arguments are developed in the text; the operator consequence of the unconditional-series theorem is also derived explicitly.
Reading route
Start with Section 2 for evaluation functionals, duality, linearization and the free space of a subset. Section 3 gives the summable-test principle and cutoff estimates. Section 4 proves compact reduction for weakly null sequences in the free space over a complete pointed metric space. Section 5 contains the compact convex Hilbert-space argument, with the approximation, selection and damping estimates written out. Section 6 gives the transfer to an arbitrary real Hilbert space. Readers already familiar with Lipschitz-free spaces may begin with Section 3 and use Section 2 as a reference. Section 1.2 records the scope and sources.
Materials
The 45-page PDF contains the complete exposition, a linked table of contents and eight references.
Suggested reference
Lipschitz-Free Spaces over Hilbert Spaces. Expository Note, Version 1, published 25 October 2026, prepared 24 October 2026. Publication responsibility: Ryotaro Tanaka. Version 1 PDF.
Responsibility and corrections
Publication responsibility: Ryotaro Tanaka.
Publication responsibility covers release, versioning and corrections; it does not assert historical priority for this proof arrangement.
This manuscript was drafted and revised using ChatGPT. Corrections should identify the version and the relevant section, equation or statement. Any correction adopted for a later version should be recorded with that version.
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