no. 3
Some embedding properties of Hilbert subspaces in topological vector spaces
Annales de l'Institut Fourier, Tome 21 (1971) no. 3, pp. 1-12.

On démontre un théorème général sur les sous-espaces hilbertiens des espaces dualement nucléaires. Tous les résultats antécédents de K. Maurin et de l’auteur sur la régularité des fonctions propres généralisées sont des corollaires simples du théorème obtenu. En plus on donne quelques suppléments aux travaux de L. Schwartz sur les sous-espaces hilbertiens des espaces de fonctions “régulières”.

A general theorem on Hilbert subspaces of dually nuclear spaces is proved, from which all previous results of K. Maurin and the writer on regularity of generalized eigenfunctions follow as simple corollaries. In addition some supplements to L. Schwartz’s work on Hilbert subspaces in spaces of smooth functions are given.

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     title = {Some embedding properties of {Hilbert} subspaces in topological vector spaces},
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Gerlach, Eberhard. Some embedding properties of Hilbert subspaces in topological vector spaces. Annales de l'Institut Fourier, Tome 21 (1971) no. 3, pp. 1-12. doi : 10.5802/aif.377. https://aif.centre-mersenne.org/articles/10.5802/aif.377/

[1] N. Aronszajn, Private communication.

[2] Heinz Bauer, Harmonische Räume und ihre Potential theorie, (Lecture Notes Math., N° 22). Springer-Verlag, Berlin, 1966. | Zbl

[3] E. Gerlach, On spectral representation for selfadjoint operators, Expansion in generalized eigenelements, Ann. Inst. Fourier (Grenoble) 15 (1965), fasc. 2, 537-574. | Numdam | MR | Zbl

[4] E. Gerlach, On the analyticity of generalized eigenfunctions (case of real variables), Ibid. 18 (1968), fasc. 2, 11-16. | Numdam | MR | Zbl

[5] E. Gerlach, Mean value properties of generalised eigenfunctions, Proceedings Edinburgh Math. Soc. (2) 17 (1970), 155-158. | Zbl

[6] A. Martineau, Sur la topologie des espaces de fonctions holomorphes, Math. Ann. 163 (1966), 62-88. | MR | Zbl

[7] K. Maurin, Analyticity of generalized eigenfunctions, Bull. Acad. Polon. Sci., Sér. sci. math. astr. phys. 14 (1966), 685-687. | MR | Zbl

[8] K. Maurin, General eigenfunction expansions on harmonic spaces, Existence of reproducing kernels, Ibid. 15 (1967), 503-507. | MR | Zbl

[9] K. Maurin, Holomorphicity of generalized eigenfunctions on complex spaces, Ibid. 15 (1967), 607-610. | MR | Zbl

[10] A. Pietsch, Nukleare lokalkonvexe Räume, Akademie-Verlag, Berlin, 1965. | Zbl

[11] L. Schwartz, Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés, (Noyaux reproduisants), J. Analyse Math. 13 (1964), 115-256. | MR | Zbl

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