Universität Konstanz
Fachbereich Mathematik und Statistik Dr. Maria Infusino
Patrick Michalski
TOPOLOGICAL VECTOR SPACES II–WS 2019/2020 Interactive Sheet
Let X be a non-trivial vector space, d:X×X →X a translation invariant metric andτd the topology induced byd.
Let us show together that (X, τd) is a metric space but not necessarily a t.v.s..
1) Show that the additiona:X×X →X isτd−continuous.
2) Let us look at a counterexample showing that the scalar multiplication m:K×X →X is not necessarilyτd−continuous.
a) Letdbe the discrete metric onXand suppose that the scalar multiplication isτd−continuous.
Then for anyx6= 0inX we have that n1x→. . . .asn→ ∞,
b) namely, for anyUneighbourhood of the origin in(X, τd)we have that. . . .
c) In particular, forU ={o} we get. . . . d) Thenx= 0, which yields a contradiction.
Hence, for the discrete metric d on X the scalar multiplication is not τd−continuous and so (X, τd) is not a t.v.s..
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