# Reflexive space

A Banach space that coincides under the canonical imbedding with its second dual (cf. Adjoint space). More precisely, let be the space dual to , i.e. the set of all continuous linear functionals defined on . If is the value of the functional on an element , then with a fixed and running through , the formula defines a linear functional on , that is, an element of the space . Let be the set of such functionals. The correspondence is an isomorphism which does not change the norm: . If , then the space is called reflexive. The spaces and , , are reflexive, and the space is not reflexive.

A space is reflexive if and only if the space is reflexive. Another criteria of reflexivity of a Banach space is weak compactness (cf. Weak topology) of the unit ball of this space.

A reflexive space is weakly complete and a closed subspace of a reflexive space is reflexive.

The concept of reflexivity naturally extends to locally convex spaces (cf. Locally convex space).

#### References

[1] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) |

[2] | K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 |

[3] | L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian) |

#### Comments

#### References

[a1] | B. Beauzamy, "Introduction to Banach spaces and their geometry" , North-Holland (1982) |

[a2] | M.M. Day, "Normed linear spaces" , Springer (1973) |

[a3] | D. van Dulst, "Reflexive and superreflexive Banach spaces" , MC Tracts , 102 , Math. Centre (1978) |

**How to Cite This Entry:**

Reflexive space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Reflexive_space&oldid=17039