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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
4
votes
Topology on the dual of a Frechet space
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 …
3
votes
Reference Request: Finite dimensional submanifolds of the space of smooth mappings
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 …
3
votes
1
answer
127
views
Openness of invertibility in Fréchet spaces for families parameterized by compact spaces
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, …
1
vote
1
answer
277
views
Cotangent bundle in the category of locally convex spaces
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 …
1
vote
Accepted
Openness of invertibility in Fréchet spaces for families parameterized by compact spaces
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 …
7
votes
2
answers
624
views
Inverse of partial differential operator as a smooth tame map
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 …
17
votes
1
answer
1k
views
Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)
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
…
16
votes
1
answer
900
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
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 …
8
votes
1
answer
501
views
Examples of topologies compatible with a given dual pair
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 …
5
votes
0
answers
140
views
Extension of elliptic complex to an exact sequence
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 …
5
votes
Intuition for failure of Implicit Function theorem on Frechet Manifolds
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 …
7
votes
Accepted
Equivariant implicit function theorem
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 …
14
votes
2
answers
521
views
Reference Request: Elliptic differential operators in the Fréchet setting
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 …
2
votes
Accepted
Reference Request: Elliptic differential operators in the Fréchet setting
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 …
13
votes
3
answers
2k
views
Space of sections of a fibre bundle with non-compact base space
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 …