6

There is a famous linear metric space constructed by R. Cauty [Un espace métrique linéaire qui n'est pas un rétracte absolu, Fund. Math. 146 (1994)] whose completion is a separable $F$-space which is not an AR. I do not have Cauty's paper handy but the latter fact is stated on the first page of Cauty's space enhanced in
[Topology Appl. 159 (2012), no. 1,...

answered Aug 25 '17 at 11:26

Igor Belegradek

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5

True: via a local chart we can assume $F^{-1}(0)$ is a closed linear subspace $N$ of $X$, and since $F_{|N}=0$, we also have $N\subset \text{ker} DF(x) $.
(A formal explanation of the latter: if we denote $i_N:N\to X$ the (bounded, linear) inclusion map, $F_{|N}=F\circ i_N:N\to Y$ is the null map and by the chain rule $0=D(F_{|N})(x)= D(F\circ i_N)(x) = DF(...

answered Sep 28 '20 at 13:19

Pietro Majer

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Assuming that $M$ and $N$ are Riemannian manifolds, the space $L^p(M,N)$ consists of measurable mappings $f:M\to N$ such that $x\to d(y_0,f(x))$ belongs to $L^p(M)$. There is no problem with this definition if the measure of $M$ is finite and a small problem if the measure of $M$ is infinite. Indeed, in the later case the constant mapping $f(x)=y_0$ belongs ...

answered Apr 17 '18 at 14:23

Piotr Hajlasz

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4

Let $\alpha\in B(H)'$ be a bounded linear functional on the space of all bounded linear operators on Hilbert space which vanishes on the subspace of compact operators $K(H)\supset H\otimes H'$. Then
$f\mapsto \alpha(d^2f(0))$ for $f\in C^\infty(H)$ is an operational tangent vector with this property, since it is a derivation:
\begin{align}
\alpha(d^2(f.g)(0)...

2

As the question is general and observes the wide of the subject; I have chosen some reference.
Path Integrals on a Compact Manifold with Non-negative Curvature
Foundations of the Theory of Semilinear Stochastic Partial Differential Equations
Stochastic differential equations on manifolds

dg.differential-geometry stochastic-processes stochastic-calculus stochastic-differential-equations hilbert-manifolds

2

I believe, the answer is (essentially) contained in the main theorem of the paper Equivalences of smooth and continuous principal bundles with infinite-dimensional structure group by Christoph Müller and Christoph Wockel:
Let $K$ be a Lie group, modeled on a locally convex space, and $M$ a finite-dimensional paracompact manifold with corners. Then each ...

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