# Topologies and bornologies determined by operator ideals, II

Studia Mathematica (1994)

- Volume: 111, Issue: 2, page 153-162
- ISSN: 0039-3223

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topWong, Ngai-Ching. "Topologies and bornologies determined by operator ideals, II." Studia Mathematica 111.2 (1994): 153-162. <http://eudml.org/doc/216125>.

@article{Wong1994,

abstract = {Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if $Q̃_p ∈ ^\{inj\}(X,X̃_p)$, where $X̃_p$ is the completion of the normed space $X_p = X/p^\{-1\}(0)$ and $Q̃_p$ 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 $Q̃_\{pq\} : X̃_q → X̃_p$ belongs to $(X̃_q,X̃_p)$. 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 seminorm p of X is -continuous if and only if every continuous seminorm p of X is a Groth()-seminorm. In this paper, we extend this equivalence to arbitrary operator ideals and discuss several aspects of these constructions which were initiated by A. Grothendieck and D. Randtke, respectively. A bornological version of the theory is also obtained.},

author = {Wong, Ngai-Ching},

journal = {Studia Mathematica},

keywords = {operator ideals; locally convex spaces; topologies; bornologies; Grothendieck spaces; ideal of absolutely summing operators; nuclear space; operator ideal},

language = {eng},

number = {2},

pages = {153-162},

title = {Topologies and bornologies determined by operator ideals, II},

url = {http://eudml.org/doc/216125},

volume = {111},

year = {1994},

}

TY - JOUR

AU - Wong, Ngai-Ching

TI - Topologies and bornologies determined by operator ideals, II

JO - Studia Mathematica

PY - 1994

VL - 111

IS - 2

SP - 153

EP - 162

AB - Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if $Q̃_p ∈ ^{inj}(X,X̃_p)$, where $X̃_p$ is the completion of the normed space $X_p = X/p^{-1}(0)$ and $Q̃_p$ 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 $Q̃_{pq} : X̃_q → X̃_p$ belongs to $(X̃_q,X̃_p)$. 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 seminorm p of X is -continuous if and only if every continuous seminorm p of X is a Groth()-seminorm. In this paper, we extend this equivalence to arbitrary operator ideals and discuss several aspects of these constructions which were initiated by A. Grothendieck and D. Randtke, respectively. A bornological version of the theory is also obtained.

LA - eng

KW - operator ideals; locally convex spaces; topologies; bornologies; Grothendieck spaces; ideal of absolutely summing operators; nuclear space; operator ideal

UR - http://eudml.org/doc/216125

ER -

## References

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