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46 views

Complex analytic set in infinite dimension

Let $E$ and $F$ be (infinite dimensional) complex Banach spaces and $f : E \rightarrow F$ a holomorphic map. Set $M = f^{-1}(\{0\})$. I have found the following results in the literature : 1) if no ...
0
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0answers
139 views

Is it possible to extend a diffeomorphism of $[0,1]^n$ to a diffeomorphism of a compact infinite-dimensional manifold?

Is it possible to extend a diffeomorphism of $[0,1]^n$ to a diffeomorphism of a compact infinite-dimensional manifold? For example, we can always extend a diffeomorphism $f$ of $[0,1]^n$ to a ...
2
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1answer
332 views

Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space?

Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$. Question: Is the space ...
2
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1answer
142 views

Manifold structure for the set of solutions to a first order elliptic system?

Consider a bounded domain $S\subset R^2$ and an elliptic system of two first order PDEs, namely a generalization of the Cauchy-Riemann system allowing nonconstant coefficients and lower order terms. ...
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4answers
928 views

How are infinite-dimensional manifolds most commonly treated?

I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fr├ęchet manifolds. Nevertheless, I am still looking for ...
4
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1answer
277 views

Banach Manifold

Let $M$ and $N$ be closed manifolds. Is it true that $C^{k}(N,M)$, which is the space of functions $f: N\to M$ such that $f\in C^{k}$, is a $C^{\infty}$ Banach manifold? If so, can you help me to ...
18
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1answer
2k views

Proposed Counterexample to a Theorem of Differential Geometry (on Banach manifolds)

This question stems from Jeff Rubin's earlier MO question and a follow-up that I posted. The former recalls the following result proved by both Serge Lang (Fundamentals of Differential Geometry, ...
21
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2answers
1k views

Non-regular Connected Hausdorff Banach Manifold

After reading this MO post, I am wondering: Is every (connected) Hausdorff Banach manifold a regular space? Though unjustified, page 53 of this paper nonchalantly states: "Note that a Hausdorff ...
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2answers
959 views

Does it make sense to talk about smooth bundles of Hilbert spaces?

Is there a notion of "smooth bundle of Hilbert spaces" (the base is a smooth finite dimensional manifold, and the fibers are Hilbert spaces) such that: 1• A smooth bundle of Hilbert spaces ...
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2answers
823 views

Are Banach Manifolds intrinsically interesting?

In the introduction to 'A convenient setting for Global Analysis', Michor & Kriegl make this claim: "The study of Banach manifolds per se is not very interesting, since they turnout to be open ...
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1answer
643 views

Is there an Error on pg. 17 of Tromba's “Teichmuller Theory in Riemannian Geometry”?

I'm pretty sure that this is a minor error, but I could use some help here. So the book I'm referring to in the title is this book (MR1164870). On pg. 16-17, he is proving that the space of almost ...
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3answers
351 views

Two notions of tangent vector for a Fréchet manifold

Let $X$ be a Frechet or Banach manifold. We can define tangent vectors by equivalence classes of smooth curves. But, we could also define them as derivations of germs of smooth functions. Do these two ...