On some topological and geometrical properties of Frechet--Hilbert spaces

D. N. Zarnadze

Izvestiya. Mathematics , Tome 41 (1993) no. 2, pp. 273-288.

Résumé

This paper contains a thorough investigation of topological, geometrical, and structural properties of Frechet spaces representable as a strict projective limit of a sequence of Hilbert spaces, and also of their strong duals, which are representable as a strict inductive limit of a sequence of Hilbert spaces. With the help of families of these spaces, representations are given for the topologies of strict inductive limits of nuclear Frechet spaces and their strong duals. In particular, these results are applicable for representing the topologies of the space $\mathscr D$ of test functions and the space $\mathscr D'$ of generalized functions.

Bibliographie

[1] Zarnadze D. N., “O predstavleniyakh topologii prostranstv osnovnykh i obobschennykh funktsii”, Trudy mezhd. konferents. po “Obobschennym funktsiyam” (Moskva, 24–28 noyabrya 1980 g.), MIAN SSSR, 1981, 250–256 | MR

[2] Kramar E., “Locally convex topological vector spaces with hilbertian seminorms”, Rev. Roum. Math. Pures et Appl., XXVI:1 (1981), 55–63 | MR

[3] Kramar E., “Linear Operations in $H$-locally convex spaces”, Rev. Roum. Math. Pures et Appl., XXVI:1 (1981), 63–79 | MR

[4] Zarnadze D. N., O nailuchshikh priblizheniyakh v lokalno vypuklykh metricheskikh vektornykh prostranstvakh, Dis. $\dots$ kand. fiz.-mat. nauk, Tbilisi, 1975

[5] Vakhaniya D. N., Chobanyan S. A., “O zadache nailuchshego priblizheniya v prostranstve vektornykh funktsii”, DAN SSSR, 264:1 (1982), 24–27 | MR

[6] Chobanian S. A., Vakhania N. N., “The linear prediction and approximation of weak second order elements”, Prediction theory and harmonic Analysis, The P. Masani Volume, North-Holland Publ. Comp., 1983

[7] Zarnadze D. N., “Refleksivnost i nailuchshie priblizheniya v prostranstvakh Freshe”, Izv. AN SSSR. Ser. matem., 44:4 (1980), 821–830 | MR | Zbl

[8] Zarnadze D. I., “O strogo pravilnykh prostranstvakh Freshe”, Matem. zametki, 31:6 (1982), 899–908 | MR | Zbl

[9] Dierolf S., Zarnadze D. N., “A note on strictly regular Frechet spaces”, Arch. Math., 42 (1984), 549–556 | DOI | MR | Zbl

[10] Zarnadze D. N., “O prostranstvakh Freshe s nekotorymi klassami proksiminalnykh podprostranstv”, Izv. AN SSSR. Ser. matem., 50:4 (1986), 711–726 | MR

[11] Bellenot S. F., Dubinsky E., “Frechet spaces without nuclear Köthe quotients”, Trans. Amer. Math. Soc., 273 (1982), 579–594 | DOI | MR | Zbl

[12] Moscatelli V. B., “Frechet spaces without continuous norms and without bases”, Bull. London Math. Soc., 12 (1980), 63–66 | DOI | MR | Zbl

[13] Behrend E., Dierolf S., Harmand P., “On a problem of Bellenot–Dybinsky”, Math. Ann., 275 (1986), 337–339 | DOI | MR | Zbl

[14] Metafune G., Moscatelli V. B., “Complemented Subspaces of Sums and Products of Banach Spaces”, Annali di Mat. Pura ed Applicata (4), 153 (1988), 175–190 | DOI | MR | Zbl

[15] Moscatelli V. B., Floret K., “On bases in strict inductive and projective limits of locally convex spaces”, Pac. Journ. Math., 119:1 (1985), 103–113 | MR | Zbl

[16] Vladimirov V. S., Obobschennye funktsii v matematicheskoi fizike, Nauka, M., 1979 | MR

[17] Zarnadze D. N., “O prostranstvakh Freshe, kazhdoe zamknutoe podprostranstvo kotorykh proksiminalno”, Tr. IVM AN GSSR, XXIX, 1990, 123–138 | MR

[18] Zarnadze D. N., “O strogo pravilnykh prostranstvakh dvoinykh posledovatelnostei”, Sb. dokl. konf. mol. uchenykh, IPM, TGU, 1983 | MR

[19] Lindenstrauss J., Safriri L., “On the complemented subspaces problem”, Israel Journ. of Math., 9 (1971), 263–279 | DOI | MR

[20] Grothendieck A., “Produits tensoriele topologiques et Espaces nucleares”, Memoirs of the AMS, 16, 1955 | MR

[21] Eberhardt V., “Beispiele topologischer Vektorräume mit der Komplementerraumeigenschaft”, Arch. Math., 26:6 (1975), 627–637 | DOI | MR

[22] Dierolf S., Floret K., “Über die Fortzetzbarkeit stetiger Normen”, Arch. Math., 35 (1980), 149–154 | DOI | MR | Zbl

[23] Grothendieck A., “Sur les espaces $(\mathscr{F})$ et $(\mathscr{DF})$”, Summa Brasil Math., 3 ; Matematika, 2:3 (1958), 81–127 | MR | MR

[24] Zarnadze D. N., “Strogaya pravilnost separabelnogo prostranstva Freshe, ne obladayuschego yadernym faktorprostranstvom Kete”, Matem. zametki, 43:1 (1988), 115–124 | MR | Zbl

[25] Albinus G., “Normartige Metriken auf metrisierbaren lokalkonvexen topologischen Vektorräumen”, Math. Nachr., 37 (1968), 177–196 | DOI | MR | Zbl

[26] Albinus G., “Approximationstheorie im Raum $C(R)$”, Beiträge zur Analysis, 3 (1970), 31–44 | MR

[27] Zarnadze D. N., “Zamechaniya o teoreme metrizatsii lineinogo topologicheskogo prostranstva”, Matem. zametki, 37 (1985), 763–773 | MR | Zbl

[28] Shefer X., Topologicheskie vektornye prostranstva, Mir, M., 1971 | MR

[29] Köthe G., Topological vector spaces, I, Springer-Verlag, Berlin, Heidelberg, New York, 1983 | MR

[30] Vladimirov V. S., “Obobschennye funktsii nad polem $p$-aditicheskikh chisel”, UMN, 43:5(263) (1988), 17–53 | MR | Zbl