Banach spaces, named after [Stefan Banach]? who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions. |

Banach spaces, named after [Stefan Banach]? who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions. |

for all φ in V'. As a consequence of the Hahn-Banach theorem, this map is injective; if it is also surjective, then the Banach space V is called reflexive. Reflexive spaces have many important geometric properties. |

for all φ in V'. As a consequence of the Hahn-Banach theorem, this map is injective; if it is also surjective, then the Banach space V is called reflexive. Reflexive spaces have many important geometric properties. A space is reflexive if and only if its dual is reflexive. |