Questions tagged [banach-manifold]
The banach-manifold tag has no usage guidance.
19 questions with no upvoted or accepted answers
10
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Does the continuous mapping space between topological manifolds always admit a Banach manifold structure?
Let $M$ and $N$ be smooth, i.e. $C^\infty$, manifolds. Suppose that $M$ is compact. Then for every $k \geq 0$ it is well known that $$C^k(M,N)$$ admits the structure of a smooth Banach manifold. I am ...
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493
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Reference for the Banach Manifold structure of $C^k(M,N)$
I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:
Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^...
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Global analysis on punctured surfaces
Global analysis on open manifolds seems pretty hard. For one, the space of $C^{n,\alpha}$ functions on an open manifold need not be a tame Fréchet space (see the post Are smooth functions tame? for ...
- 163
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The structure of Banach manifolds in symplectic geometry
Let $M$ be a symplectic manifold, and let $L_0$ and $L_1$ be Lagrangian submanifolds which transverse to each other. In Floer theory, we need to consider a Banach manifold $\mathcal B$ of maps $u:\...
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Sobolev space of maps between manifolds with boundary
Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds with non-empty smooth boundary.
If we consider the Sobolev space $W^{1,p}(M,N)$, is there a reference
on how to model this as a manifold?
If ...
- 3,423
4
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0
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232
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Fredholm transversality
Let $M,N$ be two Banach or Hilbert manifolds. Assume that $P\subset N$ is a smooth submanifold of $N$.
We say that a smooth map $f:M \to N$ is Fredholm transversal to $P$ if for every $x\in M$ with $f(...
- 356
4
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0
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602
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Intuition: Smooth functions on Banach Spaces
On finite-dimensional vector spaces, we all have a reasonable idea of which functions are likely to be $C^1$ or smooth. When it comes to differentiation on Banach spaces, I find that my `intuition' ...
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230
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Sobolev spaces of maps between manifolds and the Palais-Smale Condition
I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts:
Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean space....
- 139
3
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0
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94
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Are U(H) and PU(H) locally uniform topological groups with the norm topology? Towards an instance of infinite-dimensional Hilbert's Fifth Problem
In looking at the work of Enflo generalising Hilbert's Fifth Problem from the Euclidean to the Banach case, there are the following conditions:
the multiplication in the topological group is locally ...
- 35.4k
3
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245
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Regularity of the dependence of the flow on the vector field definining it
Let $M$ be a smooth compact manifold and $k \geqslant 1$.
Define $\mathfrak{X}^k(M,TM)$ to be the set of vector fields $M \rightarrow TM$ of class $C^k$. As $M$ is compact, endowing $\mathfrak{X}^k(M,...
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Leray-Schauder degree in Banach manifolds
The so called Leray-Schauder degree is usually defined for maps of the form $I - f$, where $f: X \to X$ is a compact map defined on a Banach space. Is there an extended definition for the setting of ...
- 2,095
2
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138
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Smooth derivations of a Banach space
Let $E$ be a real (or complex) Banach space. By $C^\infty(E) $ we mean the space of all functions $f:E\to \mathbb{R}(f:E\to \mathbb{C})$ which are smooth in the sense of Frechet diffetentiability. A ...
- 356
2
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171
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Second Countability hypothesis for a Banach manifold
Is the second countability hypothesis necessary to rigoruosly define a Banach Manifold (say in infinite dimension)?
In the finite-dimensional theory of manifolds, that request is included in the ...
- 21
2
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380
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Structure of $C^k$ ($k<\infty$)Riemannian metrics on a manifold
$M$ is a smooth manifold. It's known that if $M$ is compact, then the space of smooth Riemannian metrics has a Frechet manifold structure. For the space of $C^k$($k<\infty$) Riemannian metrics, ...
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Banach tori: classification up to Fréchet homeomorphisms
Consider the set $T$ in $l_p$ defined as closure of
\begin{equation}
T = \{ (x_1,\dotsc,x_n,\dotsc): x_j = \frac{1}{2^{(j/p)}} e^{it_j}, t_j \in \mathbb{R}/\mathbb{Z} \}.
\end{equation}
This seems to ...
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1
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Banach manifold structure on the moduli space of hybrid trajectories
I am reading the paper "On the Floer homology of cotangent bundles", (arXiv link) , by Abbondandolo and Schwarz and in page $35$ to define the isomorphism between the Morse complex and the ...
- 791
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94
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Infinite dimensional smooth projective geometry
Are there two infinite dimensional (Banach or Hilbert) manifolds $(P,L)$ which satisfy the axioms of a smooth projective geometry desribed in this page: Smooth Projectove Geometry
- 356
1
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0
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116
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Projective tensor product continuous?
For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V)$ denotes the set of all linear bounded endomorphisms with operator ...
- 1,813
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Dual of isometric copies into dual Banach spaces
Let $X$ be a Banach space and $X_1\xrightarrow{}X$ isometrically. Under some assumption can we guarantee that $X^*$ contains an isometric copy of $X_1^*$. I am also interested to know if this happens ...
- 1,879