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Part of the definition of a sc Banach space $E=E_0 \supset E_1 \supset \cdots$ is that $E_\infty = \bigcap_m E_m$ is dense in each $E_m$. Unfortunately, this rules out scales of Hölder spaces $E_m=C^{m,\alpha}$, since $C^\infty$ is not dense in $C^{m,\alpha}$.

I am wondering: How important is the density requirement in the theory of polyfolds? Do the implicit function theorem and the perturbation theory still work? What, if anything, does break down?

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Preliminary response...

My understanding is that, strictly speaking, everything breaks down because sc-Banach spaces are defined (with the density requirement) in the first several pages of HWZ's first polyfold paper, and the hundreds of pages that follow are all predicated on that definition. In other words, it seems to me that much of the theory would have to change in order to accommodate removing the density requirement while still guaranteeing a coherent theory with an abstract perturbation theorem that has non-trivial applications to a variety of moduli problems. For instance, one simple concern is that without that condition, it's plausible that the infinity level $E_\infty$ could be empty. Another concern is that the following would be an sc-Banach space

$\mathbb{R}^5 \supset \mathbb{R}^4 \supset \mathbb{R}^3 \supset \mathbb{R}^2 \supset \mathbb{R}^1 \supset \{0\} \supset \{0\} \supset \{0\} \supset \cdots$

This latter example could be particularly bad, since a feature of the current definition is that in finite dimensions sc-Banach spaces are essentially just $\mathbb{R}^n$ and the sc-calculus on such spaces agrees with the classical calculus. This in turn could be problematic for guaranteeing that the zero set of a (polyfold) Fredholm section has the differentiable structure of a finite dimensional manifold (or orbifold).

However, if you are simply interested in building a sc-Banach manifold or (M-)polyfold with Hölder-type spaces rather than Sobolev spaces, the following might work. Define the levels of your sc-Banach space to be the Banach subspaces given by the completion of $C^\infty$ in each Hölder norm. Density is then automatic. The inclusion maps are continuous because they are the restriction of a continuous map. And the inclusion maps are compact because the usual Hölder inclusions are compact and the subspaces of interest are closed. This should yield a Hölder-type sc-Banach space.

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