The tag has no wiki summary.

learn more… | top users | synonyms

4
votes
2answers
351 views

Infinite dimensional Riemannian geometry

My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me ...
7
votes
1answer
189 views

Monograph or rich survey on infinite-dimensional Riemann manifolds

I'm working with the space of smooth curves $\mathcal{C}$ in a smooth manifold $M$, having (different, pre-determined) fixed endpoints. I'd like to endow it with a Riemann structure (I already have a ...
1
vote
0answers
136 views

Does the coordinate ring of affine variety admit a structure of infinite dimensional variety?

We work in the category of algebraic varieties over some algebraically closed field $k$. By infinite dimensional variety I mean a filtration: $$ V_0\subset V_1\subset V_2\subset\ldots $$ where each ...
-3
votes
2answers
355 views

Are there examples of compact infinite dimensional manifolds? [closed]

Are there known examples of compact infinite dimensional manifolds? The word "manifold" is important.
0
votes
0answers
139 views

Is it possible to extend a diffeomorphism of $[0,1]^n$ to a diffeomorphism of a compact infinite-dimensional manifold?

Is it possible to extend a diffeomorphism of $[0,1]^n$ to a diffeomorphism of a compact infinite-dimensional manifold? For example, we can always extend a diffeomorphism $f$ of $[0,1]^n$ to a ...
4
votes
2answers
264 views

Can we define smooth diffeomorphisms on the Hilbert cube $[0,1]^\mathbb{N}$ ? Has it been done in the literature?

Can we define smooth diffeomorphisms on the Hilbert cube$[0,1]^\mathbb{N}$ ? Has it been done in the literature? For example, let $E$ be the set of bijections $f$ of the Hilbert cube, such that ...
3
votes
1answer
465 views

Is it possible to define a structure of differentiable manifold on the Hilbert cube $[0,1]^\mathbb{N}$?

Is it possible to define a structure of differentiable (smooth) manifold on the Hilbert cube $[0,1]^\mathbb{N}$ ? Has it been done in the literature? In textbooks, only the Banach case is treated, ...
6
votes
2answers
463 views

Homotopy problem for infinite dimensional topological space II

This post here is a specification of this post. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of intrinsic metric spaces verifying : $X_{n}$ have topological dimension $n$. $X_{n+1}$ is ...
0
votes
2answers
272 views

Homotopy problem for infinite dimensional topological space

Let $X$ be an infinite dimensional topological space such that : $ \forall n \in \mathbb{N}$, $ \exists X_{n} \subset X$, $n$-dimensional subspaces verifying : $\forall r<n$, the homotopy ...
2
votes
1answer
332 views

Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space?

Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$. Question: Is the space ...
3
votes
2answers
231 views

Constructing a special infinite-dimensional vector bundle

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 work without this ...
4
votes
0answers
339 views

Infinite dimensional algebraic geometry

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 have a specific ...
2
votes
1answer
175 views

Coercive Symmetric Bilinear form on a Hilbert space

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 continuous symmetric ...
2
votes
0answers
164 views

Is it true that a solid, minihedral cone in infinite dimensions cannot be regular?

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_+ \cap (-V_+) = ...
2
votes
1answer
284 views

Variant of the Riemann Mapping Theorem for $Conf(\mathbb H^2)$?

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 mapping. Moreover, such ...
1
vote
1answer
125 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 ...
21
votes
4answers
929 views

How are infinite-dimensional manifolds most commonly treated?

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 am still looking for ...
2
votes
1answer
148 views

Dimension of An Ultraproduct Field as a Vector Space over Another Ultraproduct Field

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 to see that $[K : ...
2
votes
0answers
182 views

What are the current possibilities for infinite-dimensional manifolds? [closed]

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 are generalisations ...
3
votes
3answers
242 views

Is there a good metric under which a sequence of compact sets can converge to an infinite dimensional set?

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 sequence, and there is no ...
0
votes
2answers
148 views

Matrices whose range is equal to the column set [closed]

Is there such a thing? I suppose there are two cases in which they may exist: over a finite field or infinite matrices (but let's stick to countable matrices then).
4
votes
1answer
376 views

Representations of infinite dimensional Lie algebras as vector fields on manifolds

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 $l_n$ span the Lie ...
4
votes
0answers
146 views

Co-filtered and pro-finite manifolds, filtered algebras, and differential calculus on them

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 and special case I ...
1
vote
0answers
105 views

Mappings preserving convex compactness

Let $H$ be a Hilbert space. How can one describe continuous mappings $F:H \to H$ that satisfy the following condition: There exist two elements $c$, $F(c) \neq c$ and a convex compact $M$ containing ...
1
vote
1answer
625 views

Trace in an Infinite dimensional space [closed]

How do we define trace of an infinite dimensional space? How one can compute the trace of an infinite dimensional matrix?
5
votes
0answers
548 views

Hodge theory for Lie algebra (co)homology

Let $G$ be a simple Lie group and $P$ its parabolic subgroup such that on the level of Lie algebras we have $\mathfrak{p} = \mathfrak{g}_0 \oplus \mathfrak{n}$. The dual $\mathfrak{n}^{\*}$ is ...
5
votes
1answer
384 views

Is there a “Cartan product” of Harish-Chandra modules?

If $\lambda,\mu$ are two dominant weights for a Lie group $G$, then there is a canonical (up to scale, perhaps) surjection $V_\lambda \otimes V_\mu \to V_{\lambda+\mu}$ of finite-dimensional ...
4
votes
2answers
389 views

Image of the Hilbert space under a continuous bijection

Consider a continuous bijection $X\to Y$ such that $X$ is homeomorphic to a separable infinite dimensional Hilbert space. I wonder what can be said about topological properties of $Y$. To exclude ...
0
votes
2answers
408 views

$\infty$-forms and $\infty$-plectic geometry

Can you have $\infty$-forms on infinite-dimensional manifolds or elsewhere and what are they used for?
11
votes
1answer
416 views

Status of the compact AR problem?

The so-called "compact AR Problem" reads: Is every compact convex set in a metrizable topological vector space an absolute retract? It is open according to the chapter by T. Banakh, R. Cauty ...
6
votes
2answers
797 views

Fréchet manifolds vs ILH manifolds

What is the precise relation between ILH manifolds and Fréchet manifolds? Specifically: Does any ILH manifold has a canonical structure of a Fréchet manifold? If so, is it true that any ILH ...
11
votes
3answers
1k views

Hom(A,C) ⊗ Hom(B,D) injects into Hom(A⊗B,C⊗D): when? why?

Sorry for asking a linear algebra question on a research forum, but this seems to be either a case of extreme blindness on my side, or a case of a result lying much deeper than it seems. The ...
3
votes
0answers
462 views

Infinite Linear Programming

I'm trying to prove optimality for a continuous linear program. That is, I have a linear program with an uncountable number of variables and constraints. I'm not sure how to demonstrate feasibility ...
1
vote
0answers
236 views

The finite-dimensional distributions of infinte-dimensional limit of finite-dimensional vectors

Suppose we have the process $\{\varepsilon_t,t\in \mathbb{N}\}$. Suppose that this the finite-dimensional distributions of this process are Gaussian, i.e. for any $t_1,...,t_n$, vector ...
1
vote
1answer
186 views

Complement of a closed star-shaped subset in a Frechet space

Let $U$ be the complement of a closed star-shaped subset in a separable infinite-dimensional Frechet space. Since every separable Frechet space is homeomorphic to $l_2$, one knows that $U$ is a ...
1
vote
1answer
235 views

Group scheme of infinite dimensional linear groups ?

Hi there, I know there are fairly straightforward ways to write down the schemes of infinite dimensional projective spaces (not restricting myself to only countable dimensions), but what happens with ...
8
votes
1answer
451 views

Infinite Grassmannians and their coordinate rings

I'm currently thinking about some combinatorics associated to an infinite analogue of the coordinate rings of the Grassmannians $Gr(2,n)$. The combinatorics should be thought of as relating to ...
25
votes
10answers
2k views

Are infinite dimensional constructions needed to prove finite dimensional results?

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 they sometimes help me ...
20
votes
3answers
1k views

The third axiom in the definition of (infinite-dimensional) vector bundles: why?

Serge Lang's Differential and Riemannian Manifolds is a no doubt the best available reference for the theory of not-necessarily-finite-dimensional differential manifolds, but unfortunately it suffers ...
1
vote
2answers
679 views

A question about the affine Grassmanian

For $SL(2, \mathbf{C} ((t)))$ the affine Grassmanian is defined as: $$SL(2, \mathbf{C}((t))) / SL(2, \mathbf{C} [[t]])$$ Now that is fine but $SL(2, \mathbf{C} ((t)))$ has smaller parabolic subgroups. ...
8
votes
1answer
583 views

Can ⨁_I A be isomorphic to ∏_I A for infinite I?

Suppose $A$ is a non-zero ring (say commutative unital) and $I$ is an infinite set. Can it happen that there is an isomorphism of $A$-modules $\bigoplus_{i\in I}A\cong \prod_{i\in I}A$? The ...
9
votes
1answer
473 views

How manifold-like is Aut(C^n) in the holomorphic category?

This question is similar to, but not the same as this one. Take the space of automorphisms of $\mathbb{C}^n$ in the holomorphic category, with the compact-open topology. For $n=1$ this is just ...
9
votes
5answers
1k views

Ricci Curvature in Infinite Dimensions?

Is there a good notion of "Ricci curvature" in infinite dimensions? My intuitive understanding of Ricci curvature is that it is some kind of an "average" of the curvature tensor over "different ...
18
votes
3answers
2k views

Topology of function spaces?

Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds. Let ...
6
votes
3answers
1k views

(When) does Schur's lemma break?

Edit: I wrote the following question and then immediately realized an answer to it, and moonface gave the same answer in the comments. Namely, $\mathbb C(t)$, the field of rational functions of ...
8
votes
2answers
958 views

What is the infinite-dimensional-manifold structure on the space of smooth paths mod thin homotopy?

This question is motivated by the recent paper An invitation to higher gauge theory by Baez and Huerta, and the 2007 paper Parallel Transport and Functors by Schreiber and Waldorf. Let $M$ be a ...
20
votes
2answers
940 views

Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?

There are two ways to define smooth mapping spaces and I want to know how they compare? Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
5
votes
1answer
795 views

Are smooth functions on an uncountable sum continuous?

Consider the linear space $\sum_{\mathbb{R}} \mathbb{R}$. As in the Frolicher-Kriegl-Michor view, we make this into a Frolicher space as follows. Equip it with the locally convex topology of the ...
6
votes
1answer
522 views

Infinite dimensional Newlander-Nirenberg theorem

The Newlander-Nirenberg theorem states that an almost complex structure is integrable if and only if the Nijenhuis tensor vanishes. I heard that this statement is not true in infinite dimensions, ...
7
votes
3answers
954 views

Constructing Affine Kac-Moody Groups

Does anyone know a simple construction for Affine Kac-Moody groups? There is a book by Kumar ("Kac-Moody groups, their flag varieties, and representation theory") that does the construction for the ...