Note on initial topologies on rational vector spaces induced by realvalued linear mappings |
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Authors: | Peter Schroth |
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Institution: | (1) Institut für Analysis, Technische Universität Braunschweig, Pockelsstrasse 14, D-3300 Braunschweig, West Germany |
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Abstract: | LetG be a vector space over the field of rational numbers andf, g:G -linear mappings. equipped with the usual norm topology. Denote by
f
,
g
the initial topologies onG induced byf respectivelyg.Then the following result holds: If there is a nonvoid open setU whose complement contains at least one inner point such thatf
–1
U
g
, then there is ac withf=cg. In particular, iff0, the topologies coincide.Furthermore, a -linear mappingh: (G,
f
)(G,
g
) is continuous if and only if there is a real constantc withg
o
h=cf.Dedicated to Professor János Aczél on his 60th birthday |
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Keywords: | Primary 39B10 54A10 20K20 |
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