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Predual

Banach space of a dual

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Source updated March 28, 2026

In mathematics, the predual of an object D is an object P whose dual space is D.

For example, the predual of the space of bounded operators is the space of trace class operators, and the predual of the space L^∞(R) of essentially bounded functions on R is the Banach space L¹(R) of integrable functions.

In operator algebra, if a dual Banach/operator space $A$ is realized as the dual of some Banach space $A_{*}$ , then $A_{*}$ is called the predual of $A$ (Formally: $A\cong (A_{*})^{*}$ ) The predual $A_{*}$ induces a weak topology on $A$ , under which algebra operations are separately weak continuous.

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