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Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric. It is essentially …

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates …

While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why tra …

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) whic …

Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$. For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9 …

Yes, the Hodge decomposition is a topological decomposition with respect to the $C^\infty$-topology. One can argue, for example, that the Laplace-Beltrami $\Delta$ operator is elliptic and hence can b …

Let $\langle X, Y \rangle$ be a pair of vector spaces put in duality by a non-degenerate bilinear form $\langle \cdot, \cdot \rangle: X \times Y \to \mathbb{R}$. A topology $\tau$ on $X$ is called com …

Your intuition is right. To endow the space of sections of a fiber bundle $F$ with a manifold structure at $\phi \in \Gamma^\infty(F)$ you consider a tubular neighborhood (respecting the fiber structu …

The equivariant version of the implicit function theorem is the following. Let $f: \mathbb{R}^p \times \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function (possibly only defined on open neighborhoods …

Tameness for maps is one of the main ingredients for the Nash-Moser inverse function theorem. A linear map $f: X \to Y$ between Fŕechet spaces with fixed seminorms is called tame if we have an estima …

This is meant as a long comment to the very good answer by Pedro Ribeiro. There is a nice analog of the variational bicomplex in the functional framework. Namely, the space of differential forms on $M …

Yes, the spaces are isomorphic as Fréchet spaces. This is often called the exponential law and holds for every compact manifolds $M$ and $N$, $$C^\infty(M \times N) = C^\infty(M, C^\infty(N))$$ and as …

This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator. Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential op …

As Henry T. Horton already pointed out, there exists a Fréchet version of the inverse function theorem. Therefore, the easiest way to get an intuition about what can go wrong with the classical Banach …

This map is not smooth for every vector space topology on the dual space, see Remark I.3.9. in Neeb, K.-H. "Towards a Lie theory of locally convex groups" 2006 for an explicit counterexample. Thus t …