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2answers
193 views

Question on Morse inequalities

I want to understand why: From K.C Chang's book "Infinite Dimensional Morse Theory and Multiple Solution Problems": if i have then $(4.1)$ is formal : it means that EDIT1: $(4.1)$ tel us that ...
1
vote
1answer
215 views

Integration by parts wrt. a Morse function on its basin of attraction

Let $f:\mathbb R^n\to\mathbb R$ be a Morse function with uniform nondegenerate Hessian at critical points, i.e. for some $\delta>0$ $$ \forall x \in \{\nabla f=0\}\;\forall \xi\in\mathbb R^n: \quad ...
1
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1answer
255 views

Separability of the space of bounded continuous maps

Let $O$ be an open subset of the separable Hilbert space H and $k\geq0$ . Consider $C_b^k(O, Sym(H))$, the space of k-times continuously differentiable maps with values in the bounded symmetric ...
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0answers
220 views

Gradient of the energy functional in H^{1,2}-norm

I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold Q. As such, one considers H^{1,2}-maps of the circle into Q. The energy functional ...