Some embedding properties of Hilbert subspaces in topological vector spaces
Annales de l'Institut Fourier, Volume 21 (1971) no. 3, p. 1-12
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.
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”.
@article{AIF_1971__21_3_1_0,
     author = {Gerlach, Eberhard},
     title = {Some embedding properties of Hilbert subspaces in topological vector spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {21},
     number = {3},
     year = {1971},
     pages = {1-12},
     doi = {10.5802/aif.377},
     zbl = {0222.46024},
     mrnumber = {52 \#11642},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1971__21_3_1_0}
}
Gerlach, Eberhard. Some embedding properties of Hilbert subspaces in topological vector spaces. Annales de l'Institut Fourier, Volume 21 (1971) no. 3, pp. 1-12. doi : 10.5802/aif.377. http://www.numdam.org/item/AIF_1971__21_3_1_0/

[1] N. Aronszajn, Private communication.

[2] Heinz Bauer, Harmonische Räume und ihre Potential theorie, (Lecture Notes Math., N° 22). Springer-Verlag, Berlin, 1966. | Zbl 0142.38402

[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 32 #8172 | Zbl 0135.16702

[4] E. Gerlach, On the analyticity of generalized eigenfunctions (case of real variables), Ibid. 18 (1968), fasc. 2, 11-16. | Numdam | MR 52 #11641 | Zbl 0175.43501

[5] E. Gerlach, Mean value properties of generalised eigenfunctions, Proceedings Edinburgh Math. Soc. (2) 17 (1970), 155-158. | Zbl 0204.13504

[6] A. Martineau, Sur la topologie des espaces de fonctions holomorphes, Math. Ann. 163 (1966), 62-88. | MR 32 #8109 | Zbl 0138.38101

[7] K. Maurin, Analyticity of generalized eigenfunctions, Bull. Acad. Polon. Sci., Sér. sci. math. astr. phys. 14 (1966), 685-687. | MR 34 #8232 | Zbl 0167.13705

[8] K. Maurin, General eigenfunction expansions on harmonic spaces, Existence of reproducing kernels, Ibid. 15 (1967), 503-507. | MR 38 #1542 | Zbl 0167.13801

[9] K. Maurin, Holomorphicity of generalized eigenfunctions on complex spaces, Ibid. 15 (1967), 607-610. | MR 38 #562 | Zbl 0167.13802

[10] A. Pietsch, Nukleare lokalkonvexe Räume, Akademie-Verlag, Berlin, 1965. | Zbl 0152.32302

[11] L. Schwartz, Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés, (Noyaux reproduisants), J. Analyse Math. 13 (1964), 115-256. | MR 31 #3835 | Zbl 0124.06504