Banach space
Noun
Meaning
Banach space (plural Banach spaces)
(functional analysis) A normed vector space which is complete with respect to the norm, meaning that Cauchy sequences have well-defined limits that are points in the space.
Examples
A Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space.
A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space.
An infinite-dimensional Banach space X is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces.
But if there are only finitely many summands, then the Banach space direct sum is isomorphic to the Hilbert space direct sum, although the norm will be different.
Conversely, the Lindenstrauss–Tzafriri theorem asserts that if every closed subspace of a Banach space is complemented, then the Banach space is isomorphic (topologically) to a Hilbert space.
Being the dual of a normed space, the bidual X ′′ is complete, therefore, every reflexive normed space is a Banach space.