Banach manifold

Let $X$ be a set. An atlas of class $C^r,$ $r\; \backslash geq\; 0,$ on $X$ is a collection of pairs (called charts) $\backslash left(U\_i,\; \backslash varphi\_i\backslash right),$ $i\; \backslash in\; I,$ such that

1. each $U\_i$ is a subset of $X$ and the union of the $U\_i$ is the whole of $X$;
2. each $\backslash varphi\_i$ is a bijection from $U\_i$ onto an open subset $\backslash varphi\_i\backslash left(U\_i\backslash right)$ of some Banach space $E\_i,$ and for any indices $i\; \backslash text\{\; and\; \}\; j,$ $\backslash varphi\_i\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)$ is open in $E\_i;$
3. the crossover map $$\backslash varphi\_j\; \backslash circ\; \backslash varphi\_i^\{-1\}\; :\; \backslash varphi\_i\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)\; \backslash to\; \backslash varphi\_j\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)$$ is an $r$-times continuously differentiable function for every $i,\; j\; \backslash in\; I;$ that is, the $r$th Fréchet derivative $$\backslash mathrm\{d\}^r\backslash left(\backslash varphi\_j\; \backslash circ\; \backslash varphi\_i^\{-1\}\backslash right)\; :\; \backslash varphi\_i\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)\; \backslash to\; \backslash mathrm\{Lin\}\backslash left(E\_i^r;\; E\_j\backslash right)$$ exists and is a continuous function with respect to the $E\_i$-norm topology on subsets of $E\_i$ and the operator norm topology on $\backslash operatorname\{Lin\}\backslash left(E\_i^r;\; E\_j\backslash right).$

One can then show that there is a unique topology on $X$ such that each $U\_i$ is open and each $\backslash varphi\_i$ is a homeomorphism. Very often, this topological space is assumed to be a Hausdorff space, but this is not necessary from the point of view of the formal definition.

If all the Banach spaces $E\_i$ are equal to the same space $E,$ the atlas is called an $E$-atlas. However, it is not a priori necessary that the Banach spaces $E\_i$ be the same space, or even isomorphic as topological vector spaces. However, if two charts $\backslash left(U\_i,\; \backslash varphi\_i\backslash right)$ and $\backslash left(U\_j,\; \backslash varphi\_j\backslash right)$ are such that $U\_i$ and $U\_j$ have a non-empty intersection, a quick examination of the derivative of the crossover map

$$\backslash varphi\_j\; \backslash circ\; \backslash varphi\_i^\{-1\}\; :\; \backslash varphi\_i\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)\; \backslash to\; \backslash varphi\_j\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)$$

shows that $E\_i$ and $E\_j$ must indeed be isomorphic as topological vector spaces. Furthermore, the set of points $x\; \backslash in\; X$ for which there is a chart $\backslash left(U\_i,\; \backslash varphi\_i\backslash right)$ with $x$ in $U\_i$ and $E\_i$ isomorphic to a given Banach space $E$ is both open and closed. Hence, one can without loss of generality assume that, on each connected component of $X,$ the atlas is an $E$-atlas for some fixed $E.$

A new chart $(U,\; \backslash varphi)$ is called compatible with a given atlas $\backslash left\backslash \{\backslash left(U\_i,\; \backslash varphi\_i\backslash right)\; :\; i\; \backslash in\; I\backslash right\backslash \}$ if the crossover map

$$\backslash varphi\_i\; \backslash circ\; \backslash varphi^\{-1\}\; :\; \backslash varphi\backslash left(U\; \backslash cap\; U\_i\backslash right)\; \backslash to\; \backslash varphi\_i\backslash left(U\; \backslash cap\; U\_i\backslash right)$$

is an $r$-times continuously differentiable function for every $i\; \backslash in\; I.$ Two atlases are called compatible if every chart in one is compatible with the other atlas. Compatibility defines an equivalence relation on the class of all possible atlases on $X.$

A $C^r$-manifold structure on $X$ is then defined to be a choice of equivalence class of atlases on $X$ of class $C^r.$ If all the Banach spaces $E\_i$ are isomorphic as topological vector spaces (which is guaranteed to be the case if $X$ is connected), then an equivalent atlas can be found for which they are all equal to some Banach space $E.$ $X$ is then called an $E$-manifold, or one says that $X$ is modeled on $E.$

Every Banach space can be canonically identified as a Banach manifold. If $(X,\; \backslash |\backslash ,\backslash cdot\backslash ,\backslash |)$ is a Banach space, then $X$ is a Banach manifold with an atlas containing a single, globally-defined chart (the identity map).

Similarly, if $U$ is an open subset of some Banach space then $U$ is a Banach manifold. (See the classification theorem below.)

It is by no means true that a finite-dimensional manifold of dimension $n$ is {{em|globally}} homeomorphic to $\backslash Reals^n,$ or even an open subset of $\backslash Reals^n.$ However, in an infinite-dimensional setting, it is possible to classify "well-behaved" Banach manifolds up to homeomorphism quite nicely. A 1969 theorem of David Henderson{{sfn|Henderson|1969|p=}} states that every infinite-dimensional, separable, metric Banach manifold $X$ can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, $H$ (up to linear isomorphism, there is only one such space, usually identified with $\backslash ell^2$). In fact, Henderson's result is stronger: the same conclusion holds for any metric manifold modeled on a separable infinite-dimensional Fréchet space.

The embedding homeomorphism can be used as a global chart for $X.$ Thus, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space.

• {{annotated link|Banach bundle}}
• {{annotated link|Differentiation in Fréchet spaces}}
• {{annotated link|Finsler manifold}}
• {{annotated link|Fréchet manifold}}
• Global analysis – which uses Banach manifolds and other kinds of infinite-dimensional manifolds
• {{annotated link|Hilbert manifold}}

{{reflist}}

{{reflist|group=note}}

• {{cite book|last1=Abraham|first1=Ralph|last2=Marsden|first2=J. E.|last3=Ratiu|first3=Tudor|year=1988|title=Manifolds, Tensor Analysis, and Applications|publisher=Springer|location=New York|isbn=0-387-96790-7}}
• {{cite journal|last=Anderson|first=R. D.|title=Strongly negligible sets in Fréchet manifolds|journal=Bulletin of the American Mathematical Society|publisher=American Mathematical Society (AMS)|volume=75|issue=1|year=1969|issn=0273-0979|doi=10.1090/s0002-9904-1969-12146-4|pages=64–67|s2cid=34049979 |url=https://www.ams.org/journals/proc/1969-023-03/S0002-9939-1969-0248883-5/S0002-9939-1969-0248883-5.pdf}}
• {{cite journal|last1=Anderson|first1=R. D.|last2=Schori|first2=R.|title=Factors of infinite-dimensional manifolds|journal=Transactions of the American Mathematical Society|publisher=American Mathematical Society (AMS)|volume=142|year=1969|issn=0002-9947|doi=10.1090/s0002-9947-1969-0246327-5|pages=315–330|url=https://www.ams.org/journals/tran/1969-142-00/S0002-9947-1969-0246327-5/S0002-9947-1969-0246327-5.pdf}}
• {{cite journal|last=Henderson|first=David W.|year=1969|title=Infinite-dimensional manifolds are open subsets of Hilbert space|journal=Bull. Amer. Math. Soc.|volume=75|pages=759–762|doi=10.1090/S0002-9904-1969-12276-7|mr=0247634|issue=4|doi-access=free}}
• {{cite book|last=Lang|first=Serge|authorlink=Serge Lang|title=Differential manifolds|year=1972|publisher=Addison-Wesley Publishing Co., Inc.|location=Reading, Mass.–London–Don Mills, Ont.}}
• {{cite book|last=Zeidler|first=Eberhard|year=1997|title=Nonlinear functional analysis and its Applications. Vol.4|publisher=Springer-Verlag New York Inc.}}

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