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update acknowledgements
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inakleinbottle committed May 22, 2024
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part by The Alan Turing Institute under the EPSRC grant EP/N510129/1, the Data
Centric Engineering Programme (under the Lloyd’s Register Foundation grant
G0095), and the Defence and Security Programme (funded by the UK Government).
For the purpose of Open Access, the author has applied a CC-BY public
copyright licence to any Author Accepted Manuscript version arising from this
submission.
Terry Lyons was additionally supported by the Hong Kong Innovation and
Technology Commission (InnoHK project CIMDA). For the purpose of Open Access,
the author has applied a CC-BY public copyright licence to any Author Accepted
Manuscript version arising from this submission.
---

## Introduction
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In some sense, the signature is a *unversal non-linearity* on paths.
Continuous functions on the path can be approximated (uniformly) by linear
functionals ($\mathrm{G}(V)\to \mathbb{R}$) on the signature.
functionals ($\mathrm{G}TL(V)\to \mathbb{R}$) on the signature.
This is a consequence of the Stone-Weierstrass theorem.
The *dual* of $\mathrm{T}((V))$ is $\mathrm{Sh}((V)):=\mathrm{T}((V^{\ast}))
\cong\mathrm{T}((V))$
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