A sequence space is a vector space of scalar sequences, closed under coordinatewise addition and scalar multiplication. Examples include the l-p sequence space, the l-infinity sequence space and spaces of convergent sequences.
On , is a norm, bounded above by . The partial-sum map is an isometric linear embedding into the space of convergent sequences with the supremum norm. Its range contains every eventually constant sequence: take successive differences of such a sequence to obtain a finitely supported element of . These sequences are dense in , so the completion of a normed space is .
The range is proper: partial sums of the alternating harmonic series converge but their successive differences do not belong to . The finite truncations of these differences are -Cauchy and cannot have a limit in . Thus this normed version of is incomplete, despite the completeness of its usual norm.
Articles by others on the same topic
In the context of mathematics, particularly in functional analysis and topology, a **sequence space** is a type of vector space formed by sequences of elements from a given set, typically a field like the real numbers or complex numbers. A sequence space can be defined with various structures and properties, such as norms or topologies, depending on how the sequences are used or the context in which they are applied.