On countable codimensional subspaces in ultra-(DF) spaces. (English) Zbl 0597.46004
It is proved that a countable codimensional subspace G of an Ultra-(DF) space E is an Ultra-(DF) space provided G has property (b), i.e. for every bounded subset B of E the codimension of G in the linear span of \(G\cup B\) is finite. Moreover, the property of being Ultra-(DF) is also maintained in subspaces of countable codimension, if the initial Ultra- (DF) space is sequentially complete and boundedly summing, or an ultrabarrelled space.
MSC:
| 46A04 | Locally convex Fréchet spaces and (DF)-spaces |
Keywords:
countable codimensional subspace; Ultra-(DF) space; subspaces of countable codimension; ultrabarrelled spaceReferences:
| [1] | N. Adasch, B. Ernst and D. Keim,Topological vector spaces, Lecture Notes in Math.,639 (Springer, 1978). · Zbl 0397.46005 |
| [2] | N. Adasch, Lokalkonvexe Räume mit einer Fundamentalfolge beschränkter Teilmengen,Math. Ann.,199 (1972), 257–261. · Zbl 0238.46001 · doi:10.1007/BF01431479 |
| [3] | N. Adasch and B. Ernst, Teilräume gewisser topologischer Vektorräume,Collectanea Math.,24 (1973), 27–39. · Zbl 0274.46007 |
| [4] | B. Ernst, Ultra-(DF)-Raume,J. reine angew. Math.,258 (1973), 87–102. · Zbl 0231.46001 · doi:10.1515/crll.1973.258.87 |
| [5] | A. Grothendieck, Sur les espaces (F) et (DF),Summa Brasil. Math.,3 (1954), 57–123. · Zbl 0058.09803 |
| [6] | J. Kąkol, Countable condimensional subspaces of spaces with topologies determined by a family of balanced sets,Comm. Math.,23 (1983), 249–257. · Zbl 0547.46003 |
| [7] | G. Köthe,Topological vector spaces I, Springer (Berlin-Heidelberg-New York, 1969). · Zbl 0179.17001 |
| [8] | W. Ruess, Generalized inductive limit topologies and barrelledness properties,Pacif. J. Math.,63 (1976), 499–516. · Zbl 0325.46005 |
| [9] | M. Valdivia, OnDF spaces,Math. Ann.,191 (1973), 38–43. · Zbl 0204.12802 · doi:10.1007/BF01433469 |
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