Abstract
We consider upper semi-continuous compact-valued (usco) maps with values in a Banach lattice. Recently, it was shown that the space M(X, Y ) of minimal upper semi-continuous compact-valued maps from a topological space X into a metrizable topological vector space Y is a vector space which contains the space C(X, Y ) of continuous functions from X into Y as a linear subspace. In this paper, we consider the situation when the range space is a Banach lattice E. In this case, C(X, E) is a Riesz space with respect to the usual pointwise ordering.We show that M(X, E) is equipped in a natural way with a partial order that extends the order on C(X, E). With respect to this order, M(X, E) is an Archimedean Riesz space. Moreover, if E has compact order intervals, then M(X, E) is Dedekind complete. An application is made to the characterisation of the Dedekind completion of C(X, E).
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© 2016 Springer International Publishing Switzerland
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van der Walt, J.H. (2016). The Riesz Space of Minimal Usco Maps. In: de Jeu, M., de Pagter, B., van Gaans, O., Veraar, M. (eds) Ordered Structures and Applications. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-27842-1_28
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DOI: https://doi.org/10.1007/978-3-319-27842-1_28
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-27840-7
Online ISBN: 978-3-319-27842-1
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