A double look at duality

The two forms of duality that are encountered most frequently by the systems theorist-linear duality and convex duality-are examined. Linear duality has a strong algebraic characterization which extends to other structures such as groups and modules. Convex duality, on the other hand, capitalizes so strongly on the vector space structure that the resulting powerful theory (which is typically interpreted geometrically) loses the algebraic flavor of its roots. An algebraic characterization of convex duality is presented that generalizes the standard algebraic characterization of linear duality. This provides a link between the two forms of duality most important for the systems theorist. The algebraic and geometric interpretations together give a double view of duality as used in systems theory