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