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Dispersive and explosive mappings

Published online by Cambridge University Press:  09 April 2009

T. K. Sheng
Affiliation:
Faculty of Mathematics, University of Newcastle, New South Wales, 2308, Australia.
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Let Q, R be rational numbers and real numbers respectively. We use V(F) and W(F) to denote finite dimensional inner product spaces over F. Given V(Q), we use V(R) for the smallest inner space over R containing V(Q). It is known that an R-homomorphism of V(R) to W(R) is continous. We prove that if a Q-homomorphism f: V(R)W(R), then f is dispersive, i.e., given any v0V(Q) and ε > 0, the image set f[D(v0, ε)], where D(v0, ε) = [v: vV(Q), ¦v – v0¦ < ε], is not bounded. It is also shown that some Q-homomorphism f: V(Q)W(Q) can be explosive in the sense that for any v0V(Q) and ε > 0, the set f[D[v0, ε)] is dense in W(Q). As a particular case of dispersive and explosive Q-homomorphisms, we show that the algebraic number field isomorphism f: Q(a)Q(β), where f(a) = β and α ≠ β or βmacr; (βmacr; being complex conjugates of β) is explosive.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1975