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The same paper shows that the converse is true: starting from time-ordered Green functions satisfying axioms T1-T7 as in subsection I.1 one can get the Schwinger functions (Theorem 1, pp. 99, add Coro …

This is more of a long(ish) comment than an actual answer... The question(s) asked above are not the same as the one posed in the title. In the former you are referring only to closed subspaces of $\m …

As pointed by LSpice's comment to the OP, the answer to 1. is yes, for $\{e_k\ |\ k\in\mathbb{Z}^d\}$ is an orthonormal (topological) basis of $L^2(\mathbb{T}^d)$. As for 2., equation (3.33) is one of …

Yes, there is. The (Constant) Rank Theorem for Banach spaces is Theorem 2.5.15 of the book of R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications (3rd. edition, Springer …

Not always. What you are talking about is also called "convergence in the sense of sesquilinear forms", because you are taking a pointwise limit in $D\times D$ of a sequence of sesquilinear forms $\al …

I'll try to explain what Igor meant in his comments in a different way, maybe it helps. Of course, any tempered distribution is a distribution in the broader sense - more precisely, any compactly sup …

The basic idea is the one put forward by Pierre PC's comment above. More precisely, let $u,\varphi,x$ be as in the last paragraph of the OP. There is no loss of generality in assuming that $\varphi(x) …

This is easy and comes from the fact that $V$, being a Fréchet space, is barrelled, that is, the locally convex topology of $V$ coincides with its strong topology $\beta(V,V')$ (where $V'$ = topologic …

Let $M$ be a $d$-dimensional (say, Hausdorff, paracompact, connected and oriented) smooth manifold, and $K\subset M$ compact with $\mathring{K}\neq\varnothing$. M. Valdivia has shown (based on previou …

This is not really an answer but rather a long-ish comment. First of all, if $M$ is not compact, $\mathfrak{A}=C(M,\mathbb{R})$ is not really a C${}^*\!$-algebra but actually only a locally C${}^*\!$- …

$\DeclareMathOperator\Ann{Ann}\DeclareMathOperator\Tr{Tr}$My answer is somewhat complementary to Nik Weaver's, and admitedly more focused on Question 2 since I have nothing more to add to the latter r …

The "infinite tensor product" picture may be useful as a sort of concrete image of the state space of a quantum field theory, but in practice is rarely used because of the technical difficulties it br …

This is not at all a complete answer, but rather an expanded update on my above comment. I shall start with a few general considerations: If $\{f_n\ |\ n\in\mathbb{N}\}$ is any Schauder basis for eit …

I do not know if it is good form for MO to cite one's own papers when answering a question, but I will take the chance. This matter is addressed in quite a bit of detail in my joint paper with Romeo B …

Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is convenient (also called locally complete, Mackey-complete or $c^\infty$-complete) if, given …