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Topological classification of strong duals to nuclear (LF)-spaces

Taras Banakh (2000)

Studia Mathematica

We show that the strong dual X’ to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: ℝ ω , ℝ ∞ , Q × ℝ ∞ , ℝ ω × ℝ ∞ , or ( ℝ ∞ ) ω , where ℝ ∞ = l i m ℝ n and Q = [ - 1 , 1 ] ω . In particular, the Schwartz space D’ of distributions is homeomorphic to ( ℝ ∞ ) ω . As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to ℝ ∞ or to Q × ℝ ∞ . In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic either...

Topologies and bornologies determined by operator ideals, II

Ngai-Ching Wong (1994)

Studia Mathematica

Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if Q ̃ p ∈ i n j ( X , X ̃ p ) , where X ̃ p is the completion of the normed space X p = X / p - 1 ( 0 ) and Q ̃ p is the canonical map. p is said to be a Groth()- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map Q ̃ p q : X ̃ q → X ̃ p belongs to ( X ̃ q , X ̃ p ) . It is well known that when is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous...

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