a vector space with a metric that allows the computation of vector length and distance between vecto

Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vecto ...

a normed vector space

In functional analysis, a Banach space is a normed vector space that allows vector length to be computed.

a complete normed vector space such that every Cauchy sequence (with respect to the metric d(x, y) = |x - y|) in B has a limit in B.

A Banach space, B, is a complete normed vector space such that every Cauchy sequence (with respect to the metric d(x, y) = |x - y|) in B has a limit in B.

a closed linear subspace of the space of bounded sequences

a Banach space is a closed linear subspace of the space of bounded sequences, ℓ∞, and contains as a closed subspace the Banach space c0 of sequences converging to zero.

A space which is complete under the metric induced by a norm

A space which is complete under the metric induced by a norm is a Banach space.

that allows vector length to be computed

In functional analysis, a Banach space is a normed vector space that allows vector length to be computed.

a metric space that is complete

The metric also allows for a definition of limits and completeness - a metric space that is complete is known as a Banach space.

a metric space

A Banach space is in particular a metric space.

the name for complete normed vector spaces, one of the central objects of study in functional analysis

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis.

the same structure as a normed vector space

A Banach space is really the same structure as a normed vector space, a normed vector space just has some extra property – that the induced metric is complete.

functional analysis

Banach spaces play a central role in functional analysis.

Any reflexive normed space

Corollary: Any reflexive normed space is a Banach space.

a strictly convex space

The Banach space is a strictly convex space (i.e., the boundary of the unit ball B contains no line segments) if and only if δ(2) = 1, i.e., if only antipodal points (of the form x and y = −x) of the unit sphere can have distance equal to 2.

sets whose members are not numbers but complicated mathematical objects such as functions or operators

Banach spaces are sets whose members are not numbers but complicated mathematical objects such as functions or operators.

Hilbert spaces

Banach spaces are a different generalization of Hilbert spaces.

a Smith space

A Banach space is a Smith space if and only if is finite-dimensional.

an associative and commutative algebra under convolution

The Banach space is an associative and commutative algebra under convolution.

test function space for time-frequency analysis

a Banach space is for these and more properties that is a natural choice of test function space for time-frequency analysis.

an F-space with an additional requirement

In particular, a Banach space is an F-space with an additional requirement that .

a TVS equipped with a complete norm generating the topology

A Banach space is a TVS equipped with a complete norm generating the topology, so a morphism (and in particular an isomorphism) of Banach spaces should preserve the norm, whereas a continuous map is merely preserving the underlying topology.