Sequentially complete

In mathematics, specifically in topology and functional analysis, a subspace S of a topological vector space (TVS) X is said to be sequentially complete or semi-complete if every Cauchy sequence in S converges to an element in S. We call X sequentially complete if it is a sequentially complete subset of itself.

Examples

Every complete space is sequentially complete but not conversely. If a TVS is metrizable then it is complete if and only if it is sequentially complete.

See also

• Cauchy net
• Complete space
• Topological vector space
• Uniform space

References

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)