Connes distribution

Connes distribution space is a ccertain analogue of the theory of generalized functions (distributions) for functional spaces.

Let α (n)\alpha_(n), nNn\in \mathbf{N} be a sequence of real strictly positive numbers and H kH_k, k>0k\gt 0 the Sobolev space in the corresponding geometric setip (typically of certain sections of a Hermitean bundle EE on a compact Riemannian manifold MM). Let

σ= nσ n \sigma = \sum_n \sigma_n

where σ kH k n\sigma_k \in H^{\otimes n}_k. For every C>0C\gt 0 set

σ 1,C,k:=C nα(n)σ n H k n \|\sigma\|_{1,C,k} := \sum C^n \alpha(n) \| \sigma_n \|_{H^{\otimes n}_k}

Banach space Co C,kCo_{C,k} is the space of σ\sigma for which σ 1,C,k<\|\sigma\|_{1,C,k}\lt \infty. The space of Connes functionals Co := C>0,k>0Co C,kCo_{\infty -}:= \cap_{C\gt 0, k\gt 0} Co_{C,k}. The Connes distribution space Co Co_{-\infty} will be its topological dual.

The Potthoff–Streit theorem allows to define flat Feynman path integrals as distributions.

A closely related approach is that of a Hida in the theory of white noise.

Gel'fand triples are often used in spectral analysis and distribution theory on infinite-dimensional spaces: