Let $M$ and $N$ be finite dimensional smooth manifolds and $p: E \rightarrow N$ a finite rank vector bundle. $M$ can be assumed to be compact if necessary but I would prefer to wor …

I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I …

I need to show one of the two following equivalent results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance. 1) Consider a co …

Hi! I am looking for basic references about infinite dimensional algebraic geometry, in particular about the $\textrm{Proj}$ of an infinite dimensional graded commutative algebra. …

I need to show the two following results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance. Consider a continuous symmetric bi …

According to the Riemann mapping theorem it is possible to map a simply connected open subset $B \subset \mathbb C$ into any other $B' \subset \mathbb C$ by a (bi-)holomorphic mapp …

Background Consider a real Banach$^1$ space $V$. We'll call a subset $V_+ \subseteq V$ a cone if $V$ is closed, $\alpha V_+ \subseteq V_+$ for every $\alpha \geqslant 0$ and $V_ …

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 …

Suppose $F \subseteq K$ are fields with $G$ an ultrafilter on an infinite set $X$. If $F^{\ast}$ and $K^{\ast}$ represent the ultraproducts respectively of $F$ and $K$, it is easy …

(DFM)-spaces are locally convex spaces which are (DF)- and (Montel)-spaces. For the subclass of (DFS)-spaces (i.e. the inductive limits of a sequence of Banach spaces with compact …

According to wikipedia, by a theorem of Henderson '69, infinite-dimensional Frechet Manifolds embed as open subspaces of Hilbert Space. They need to be seperable & metric. They …

I have a sequence of finite-dimensional, compact sets in $L^2(\Omega)$, where $\Omega\subset \mathbf{R}^2$ is closed and bounded. The dimension grows monotonically with the sequen …

Hello! I've come across a lot of questions (and nice answers) on MO, concerning infinite-dimensional manifolds and differential calculus over them, but nothing suiting the simpler …

Infinite dimensional constructions, such as spaces of diffeomorphisms, spectra, spaces of paths, and spaces of connections, appear all over topology. I rather like them, because th …

Suppose I have e.g. the Witt algebra, $\left[l_n,l_m \right] = -(n-m)l_{n+m}$. I want to realize the $l_n$ as vector fields on some manifold. The classical example is when the …