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Uniqueness of the topology on L¹(G)

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Extremera J., Mena J., Villena A.

Studia Mathematica

2002

tom 150

nr 2

163-173

Let G be a locally compact abelian group and let X be a translation invariant linear subspace of L¹(G). If G is noncompact, then there is at most one Banach space topology on X that makes translations on X continuous. In fact, the Banach space topology on X is determined just by a single nontrivial translation in the case where the dual group Ĝ is connected. For G compact we show that the problem of determining a Banach space topology on X by considering translation operators on X is closely related to the classical problem of determining whether or not there is a discontinuous translation invariant linear functional on X. As a matter of fact L¹(G) does not carry a unique Banach space topology that makes translations continuous, but translations almost determine the Banach space topology on X. Moreover, if G is connected and compact and 1 < p < ∞, then $L^{p}(G)$ carries a unique Banach space topology that makes translations continuous.

Ograniczanie wyników

1 Studia Mathematica

1 Extremera J.

1 Mena J.

1 Villena A.

1 2002

Wyniki wyszukiwania

Uniqueness of the topology on L¹(G)