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Firstly consider the following definitions:


Definition 1: We say that $M\subseteq\mathbb{R}^n$ is a (smooth) manifold with dimension $m$ if for all $p\in M$ there's a map $\psi :V\to \mathbb{R}^n$ such that the following propositions are true:

  1. $V\subseteq \mathbb{R}^m$ is open and $\psi$ is a smooth immersion;
  2. $p\in \psi [V]$ and $\psi :V\to \psi [V]$ is a homeomorphism such that $\psi [V]=U\cap M$ in which $U\subseteq\mathbb{R}^n$ is open.

Definition 2: Let $M\subseteq\mathbb{R}^m$ and $N\subseteq\mathbb{R}^n$ be two manifolds. We say that $f:M\to N$ is a smooth map in $p\in M$ if there're an open neighborhood $V\subseteq\mathbb{R}^m$ of $p$ and a smooth map $F:V\to \mathbb{R}^n$ such that $F|_{V\cap M}=f|_{V\cap M}$. Besides, if $f:M\to N$ is a smooth map in $p$ for all $p\in M$ then we say that $f$ is a smooth map.


We can see in the book "Introduction to Differential Geometry" (written by J.W. Robbin and D.A. Salamon) that the above definitions are compatible with the usual definitions of smooth manifolds and smooth maps. With those definitions many theorems about manifolds and its proofs became simpler (as you can see in that book I mentioned). Because of this I would like similar definitions that cover Banach spaces with finite dimensions.

With that in mind I thought of the following definitions:


Definition 1': Let $E$ be a $\mathbb{R}$-Banach space with finite dimension. We say that $M\subseteq E$ is a (smooth) manifold with dimension $m$ if for all $p\in M$ there's a map $\psi :V\to E$ such that the following propositions are true:

  1. $V\subseteq\mathbb{R}^m$ and $\psi $ is a smooth immersion;
  2. $p\in \psi [V]$ and $\psi :V\to \psi [V]$ is a homeomorphism such that $\psi [V]=U\cap M$ in which $U$ is open in $E$.

Definition 2': Let $E,F$ be two $\mathbb{R}$-Banach spaces with finite dimension. Suppose that $M\subseteq E$ and $N\subseteq F$ be manifolds. We say that $f:M\to N$ is a smooth map in $p\in M$ if there're an open neighborhood $V\subseteq E$ of $p$ and a smooth map $F:V\to F$ such that $F|_{V\cap M}=f|_{V\cap M}$. Besides, if $f:M\to N$ is a smooth map in $p$ for all $p\in M$ then we say that $f$ is a smooth map.


My question: Are the last two definitions compatible with the usual definitions of manifolds? Will there be any problem if I use the previous two definitions to study manifold contained in finite dimensional Banach spaces?

By saying "compatible" I mean the following: in the case $M\subseteq \mathbb{R}^n$ we can prove, for example, that definition 1 is equivalent to the usual definition of manifolds (using charts). So I want to know if this is also true in the case $M$ is a subset of a finite dimensional Banach space.

Thank you for your attention!

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    $\begingroup$ You can do this as long as you know what a smooth map $\psi:V\rightarrow E$ is here: all its Frechet derivatives are continuous. Since all the norms in finite dimensions are equivalent and the Frechet derivatives coincide with the usual ones, the two notions are "compatible". So no, there would be no problem using that definition, though I don't really see the point. $\endgroup$
    – andres1
    Aug 8, 2021 at 10:23
  • $\begingroup$ @andres1 Do you know if that definition excludes the existence of exotic smooth structures? $\endgroup$
    – rfloc
    Aug 9, 2021 at 19:15
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    $\begingroup$ Since the notions are equivalent, it doesn't exclude the existence. But this Banach space definition is only useful if you're interested in infinite dimensional manifolds. $\endgroup$
    – andres1
    Aug 9, 2021 at 20:08

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