Predual

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 ${\displaystyle A}$ is realized as the dual of some Banach space ${\displaystyle A_{*}}$, then ${\displaystyle A_{*}}$ is called the predual of ${\displaystyle A}$ (Formally: ${\displaystyle A\cong (A_{*})^{*}}$) The predual ${\displaystyle A_{*}}$ induces a weak topology on ${\displaystyle A}$, under which algebra operations are separately weak continuous.¹