predual

A predual of a Banach space is a Banach space X such that the given space Y is (isometrically) the continuous dual of X, that is, Y ≅ X*. The space X is then said to be a predual of Y. Equivalently, Y carries the weak-* topology with respect to X.

Existence and uniqueness: Not every Banach space is a dual space, so not every Y has a predual. When Y is a dual space, it has at least one predual, but preduals need not be unique. A dual space may admit many non-equivalent preduals, yielding different weak-* topologies on Y. In contrast, certain classes of algebras, most notably von Neumann algebras, have a unique predual up to isometric isomorphism, which underpins the natural weak-* topology used in that theory.

Examples: A basic example is L1, whose dual is L∞, so L1 is a predual of L∞ (though L∞ can have other preduals as well). If K is compact, the space of regular Borel measures M(K) satisfies M(K)* ≅ C(K), so C(K) is a predual of M(K). Another example is c0, since its dual is l1, making c0 a predual of l1.

Relevance: The concept links to topologies and duality in analysis. The weak-* topology on a dual space Y, defined via its predual X, plays a central role in areas such as harmonic analysis, measure theory, and the theory of operator algebras. In the study of von Neumann algebras, the predual is particularly important because it is unique and gives the canonical weak-* topology on the algebra.