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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,...
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 $0=D(F_{|N})(x)= D(F\circ i_N)(x) = DF(... 4 Assuming that$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$belongs ... 4 Let$\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 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: LetK$be a Lie group, modeled on a locally convex space, and$M\$ a finite-dimensional paracompact manifold with corners. Then each ...