Bidual and canonical embedding

Summary

The bidual $X^{**}$ of a normed space $X$ is the dual of its continuous dual space $X^*$. The canonical embedding $J \colon X \to X^{**}$ sends a vector $x$ to the functional "evaluate at $x$". By the Hahn–Banach theorem, $J$ is a linear isometry, so every normed space sits inside its bidual. Spaces for which $J$ is onto are the reflexive spaces.