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there had been no references at Hilbert space, I have added the following, focusing on the origin and application in quantum mechanics:
John von Neumann, Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.
George Mackey, The Mathematical Foundations of Quamtum Mechanics A Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963
E. Prugovecki, Quantum mechanics in Hilbert Space. Academic Press, 1971.
added pointer to
also added a parenthetical remark to the end of the entry’s footnote here
A basic thought on the infinite tensor product of $L^2$-spaces:
The “grounded” infinite tensor product of Hilbert spaces, which is mentioned in
and discussed in
has, I’ll suggest, a nicely transparent incarnation in the case of the Hilbert space of square-integrable functions on some $(X,\mu)$, because here it is the dual of the limit that gives the infinite product $X^{\times_\infty}$:
$X^\infty \;\coloneqq\; \underset{ \underset{ \mathclap{ S \in FinSub(\mathbb{N}) } }{\longleftarrow} }{\lim} \; X^{S} \;\;\;\;\;\;\; \mapsto \;\;\;\;\;\; \underset{ \underset{ \mathclap{ S \in FinSub(\mathbb{N}) } }{\longrightarrow} }{\lim} \; L^2(X)^{\otimes_S} \;\;\;\; =: \; L^2(X)^{\otimes_\infty} \,.$Here the “groundedness” of the product is just the contravariant functoriality of $L^2(-)$ applied to the projections
$\array{ S' &\xhookrightarrow{ \;\;\;\; i \;\;\;\; }& S \\ X^{S'} &\xleftarrow{ \;\;\;\; X^i \;\;\;\;}& X^S \\ L^2\big( X^{S'} \big) &\xrightarrow{ \;\;\; (X^i)^\ast \;\;\; }& L^2\big( X^{S} \big) }$which implicitly regards each $L^2$-spaces as having as vacuum state the constant unit function.
This makes one want to say that $L^2(-)$ takes (these) limits to (these) colimits, so that
$L^2(X)^{\otimes_\infty} \;\; \simeq \;\; L^2\big( X^\infty \big) \,.$Is this the case? (Just firing off this question, maybe it’s evident once I start to really think about it…)
For what it’s worth, this is appears as Ex. 6.3.11 in
Incidentally, regarding original references on the infinite tensor product in #6:
Wikipedia here calls this notion the “incomplete-” or “Guichardet-“tensor product and references
where the notion is briefly mentioned on p. 144 (and only there, it seems), without, however, calling it anything, neither “incomplete” nor “Guichardet”.
The attribution to Guichardet must refer to
where the case of Hilbert spaces is considered explicitly in section 6, while “incomplete” seems to refer to
where something at least similar is considered around Lem. 4.1.2. (But is it the same?)
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