# Questions tagged [hilbert-manifolds]

A Hilbert manifold is a manifold based on a Hilbert space.

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### $\sigma$-product of the Hilbert cube

Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$ ("eventually&...
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### 'Closedness' for infinite-dimensional separable Hilbert manifolds

In the following content, the Hilbert space refers to the unique separable infinite-dimensional real Hilbert space $H$. Hilbert manifolds are locally modelled on this space (so they must be infinite-...
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### Exponential map on the Hilbert manifold $\Omega(M,q_0,q_1)$

Consider $M$ to be a compact manifold and consider the based loop space $\Omega(M,q_0,q_1)$ of loops of class $W^{1,2}$. It can be shown that that this has a structure of an hilbert manifold. It's ...
507 views

### Implicit function theorem with continuous dependence on parameter

Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$. Let $f:X\times P\to Y$ be a continuous map such that for any parameter $p\in P$, $f_p:= f|_{X\times \{p\}}:X\to Y$ is ...
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### Tangent space of smooth Hilbert submanifolds

Let $X, Y$ be Hilbert spaces and $F:X \rightarrow Y$ smooth. Assume that $M := F^{-1}(0) \subset X$ is a smooth submanifold. Is it true that for any $x\in M$, the tangent space $T_xM$ is a Hilbert ...
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### Almost-geodesics on a Riemannian Hilbert manifold which are still almost geodesics in some submanifold

Let $H$ be a separable infinite dimensional Hilbert space, and consider it as a Hilbert manifold in the usual way (that is, with the single chart with the identity map). It is known that there always ...