Noun
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.
A Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space. Source: Internet
A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. Source: Internet
An infinite-dimensional Banach space X is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces. Source: Internet
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. Source: Internet
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. Source: Internet
Being the dual of a normed space, the bidual X ′′ is complete, therefore, every reflexive normed space is a Banach space. Source: Internet