Search Results

This is a follow-up to this question. Let $T$ be a tempered distribution on $\mathbb{R}^d$. Then there is a multiindex $\alpha \in \mathbb{N}_0^d$, an $n \in \mathbb{N}_0$ and a bounded continuous fu …

Let $\mu$ be a centred Radon Gaussian measure on a locally convex space $X$ and $q : X \to \mathbb{R}$ a seminorm that is $\mathcal{B}(X)_\mu$-measurable, where $\mathcal{B}(X)_\mu$ is the Lebesgue co …

Suppose I have a distribution $E$ such that $\phi \ast E$ is square-integrable for all $\phi \in C_c^\infty \left( \mathbb{R}^d \right)$. Is it possible to prove that $E$ is tempered? It seems plausib …

I recently stumbled across this old publication which defines the local completion of a Hausdorff locally convex space. The construction is as follows: A Hausdorff locally convex space $E$ is locally …

It turns out, that the above is actually rather simple. Clearly, $E$ is dense in $F$ with its subspace topology from $\widetilde{E}$. Furthermore, since $E$ is Mackey, it is well-known that its comple …

I have found an answer exploiting the equivalence of Lusin and Borel measurability as stated in Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures on page 6, theorem 5: Let $H : …

$\newcommand{\real}{\mathrm{real}}$I am having trouble with understanding the axiom (OS3) in this book by Glimm and Jaffe. It defines \begin{equation} \mathcal{A} = \left \{ A(\phi) = \sum_{j = 1}^N c …

Since this has confused me multiple times, I write this answer in the hope that it might help others. First, recall that reflection positivity as formulated by Osterwalder and Schrader states that \be …

I have a Fréchet space $V$ whose topology is (if it helps) induced by a family $\mathcal{P}$ of norms - not just seminorms - and on this space I have a Borel probability measure $\nu$. Now, I would li …

Let $X$ be a vector space equipped with a norm $p$ and a seminorm $q$. Denote the completion of $X$ with respect to $p$ with $X_p$ and with respect to $p+q$ by $X_{p+q}$. Then the induced map $\iota : …

Citing from Wikipedia, A Hausdorff topological space is a Lusin space if some stronger topology makes it into a Polish space. Is there a (previously studied) analogous concept of a Hausdorff (locall …

I just read the nice exposition Fermionic Path Integral on nLab and began to wonder about some details to which references appear to be lacking. Suppose we live on Euclidean space as in the Osterwalde …

I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ with the prop …

If we consider the usual formal construction of a path integral over fields with gauge symmetries e.g as in Weinbergs "The Quantum Theory of Fields - Volume 2" the notion of gauge invariance is clear …

I am following the conventions of https://arxiv.org/abs/math-ph/9902027 Let $M$ be a differentiable manifold, $E \to M$ a vector bundle over $M$ with fibre $F$, $J^1(E)$ the rank-one jet bundle over …