as complete normed linear spaces

Section 1 defines Banach spaces as complete normed linear spaces and gives a number of examples of these.

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 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 vector space that comes with a notion of size and a notion of limits

Speaking in very rough terms, a Banach space is a vector space that comes with a notion of size and a notion of limits.

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.

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.

a metric space

A Banach space is in particular a metric space.

for a definition of limits and completeness - 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 vector space over any scalar field , Kwhich is equipped with a norm and which is complete with respect to the distance function induced by the norm

A Banach space is a vector space over any scalar field K, which is equipped with a norm and which is complete with respect to the distance function induced by the norm, that is to say, for every Cauchy sequence in , there exists an element in such that or equivalently:

a normed vector space which is also a complete metric space where the metric comes from the norm

A Banach space is a normed vector space which is also a complete metric space where the metric comes from the norm.

an element in

A Banach space is a vector space over any scalar field K, which is equipped with a norm and which is complete with respect to the distance function induced by the norm, that is to say, for every Cauchy sequence in , there exists an element in such that or equivalently: