bibkey: bargmann1961analytic authors: Valentine Bargmann year: 1961 title: “On a Hilbert space of analytic functions and an associated integral transform part I” doi: 10.1002/cpa.3160140303 url: https://doi.org/10.1002/cpa.3160140303 claim: “The analytic Fock-space realization makes the exponential-vector family total.” strata_touched:

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Analytic realization of Fock space

V. Bargmann, Communications on Pure and Applied Mathematics 14, 187–214 (1961), DOI 10.1002/cpa.3160140303.

The analytic realization sends occupation coefficients to the entire power series with factorial normalization. Orthogonality to every exponential vector annihilates that power series; coefficient uniqueness then annihilates every occupation coefficient. Applying this argument in each parameter gives totalness of the two-mode exponential family. The formal constructions use these arguments as local steps on the completed countable Hilbert sums, applying the pinned analytic-series uniqueness API.

The DOI, author, title and publication year are verified against Crossref. The full text has not been inspected; this reference attributes the classical analytic realization, rather than claiming a verbatim theorem number in the article.

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DOI and bibliographic metadata: https://doi.org/10.1002/cpa.3160140303. The full text has not been inspected; no verbatim theorem number is attributed.