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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 …

Lets assume that $U$ is a compact submanifold of $\mathbb R^k$, so that we do not need to worry about boundary conditions and things happening at infinity. Then $C^\infty(U, V)$ is a smooth Fréchet ma …

Consider the following setup. Let $K$ be a compact topological space, $X$ a Fréchet space and $T:K \times X \to X$ a continuous family of linear maps (i.e. $T$ is a continuous map and $T_k \equiv T(k, …

I'm trying to understand the definition of a differential form on $M$ in the context of Fréchet spaces or, more generally, locally convex spaces. The standard procedure defines a k-form as a map $\alp …

The statement is indeed overoptimistic. As a counterexample, consider the continuous family $T: [0,1] \times C^\infty([0, 1]) \to C^\infty([0, 1])$ of linear differential operators defined by \be …

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 …

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 …

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 …

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 …

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 …

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 …

A few bits and pieces of Fredholm theory in the locally convex (and, in particular, Fréchet) setting are discussed in the literature. The most comprehensive treatment I could find were two old article …

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 …