Table of contents |

2 Examples 3 Properties |

Suppose *X* is a Banach space. We denote by *X'* its continuous dual, i.e. the space of all continuous linear maps from *X* to the base field (**R** or **C**). This is again a Banach space, as explained in the dual space article. So we can form the *double dual* *X"*, the continuous dual of *X'*. There is a natural continuous linear transformation

*J*:*X*→*X"*

*J*(*x*)(φ) = φ(*x*) for every*x*in*X*and φ in*X'*.

All Hilbert spaces are reflexive, as are the L^{p} spaces for 1 < *p* < ∞. More generally: all uniformly convex Banach spaces are reflexive.

Every closed subspace of a reflexive space is reflexive.

The promised geometric property of reflexive spaces is the following: if *C* is a closed non-empty convex subset of the reflexive space *X*, then for every *x* in *X* there exists a *c* in *C* such that ||*x* - *c*|| minimizes the distance between *x* and points of *C*. (Note that while the minimial distance between *x* and *C* is uniquely defined by *x*, the point *c* is not.)

A space is reflexive if and only if its dual is reflexive.

A space is reflexive if and only if its unit ball is compact in the weak topology.