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...ical analysis]], most spaces which arise in practice turn out to be Banach spaces as well. == Classical Banach spaces == ...

In [[mathematics]], more specifically [[general topology]], the '''rational sequence topology''' is an example of a [[Topological space#topology|topology]] give ...''x_k''} with the property that {''x_k''} [[Limit of a sequence|converge]]s to ''x'' with respect to the [[Euclidean topology]]. ...

...or a general class of spaces, similarly to weak convergence, every bounded sequence has a Delta-convergent subsequence. A sequence <math>(x_k)</math> in a metric space <math>(X,d)</math> is said to be Δ-co ...

...thematics, specifically in [[order theory]] and [[functional analysis]], a sequence of positive elements <math>\left(x_i\right)_{i=1}^{\infty}</math> in a [[Or ...<math>\ell^p</math>''' if there exists some <math>z \in X</math> and some sequence <math>\left(c_i\right)_{i=1}^{\infty}</math> in <math>\ell^p</math> such th ...

...i-barrelled''' if every [[strong dual space|strongly bounded]] (countable) sequence in <math>X^{\prime}</math> is equicontinuous.{{sfn | Khaleelulla | 1982 | p ...ally quasi-barrelled''' if every [[strong dual space|strongly]] convergent sequence in <math>X^{\prime}</math> is equicontinuous. ...

...'''σ-barrelled''' if every [[weak-* topology|weak-*]] bounded (countable) sequence in <math>X^{\prime}</math> is equicontinuous.{{sfn | Khaleelulla | 1982 | p ...''sequentially barrelled''' if every [[weak-* topology|weak-*]] convergent sequence in <math>X^{\prime}</math> is equicontinuous.{{sfn | Khaleelulla | 1982 | p ...

...f [[mathematics]], an '''ultrabarrelled space''' is a [[topological vector spaces]] (TVS) for which every ultrabarrel is a [[Neighbourhood (topology)|neighbo ...d [[balanced set|balanced]] subset of <math>X</math> and if there exists a sequence <math>\left(B_i\right)_{i=1}^{\infty}</math> of closed balanced and [[absor ...

...athematics]], a '''quasi-ultrabarrelled space''' is a [[topological vector spaces]] (TVS) for which every [[Barrelled set|bornivorous ultrabarrel]] is a [[ne ...and [[Bornivorous set|bornivorous]] subset of ''X'' and if there exists a sequence <math>\left( B_{i} \right)_{i=1}^{\infty}</math> of closed balanced and bor ...

A <math>\lambda</math>-chain between <math>x</math> and <math>y</math> is a sequence of points ...ok| last = Heinonen| first = Juha | title = Lectures on Analysis on Metric Spaces | series = Universitext | publisher = Springer-Verlag | location = New York ...

...lities more subtle than a [[convergence (mathematics)|convergence]] of a [[sequence]]. They are named after E. G. Pytkeev, who proved in 1983 that [[sequential | title = Maximally decomposable spaces ...

...espondence is implemented by mapping an operator to its [[singular value]] sequence. ...g problems about operator spaces to (more resolvable) problems on sequence spaces. ...

...mmonly used to illustrate the Schur property, is the <math>\ell_1</math> [[sequence space]]. .../math> an arbitrary member of ''X'', and <math>(x_{n})</math> an arbitrary sequence in the space. We say that ''X'' has '''Schur's property''' if <math>(x_{n}) ...

...e space denoted by '''''c''''' is the [[vector space]] of all [[convergent sequence]]s <math>\left(x_n\right)</math> of [[real number]]s or [[complex number]]s * {{annotated link|Sequence space}} ...

{{About|the vector spaces of sequences and functions|the finite-dimensional vector space distance|Che ...the latter. As a Banach space they are the continuous dual of the Banach spaces <math>\ell_1</math> of absolutely summable sequences, and <math>L^1 = L^1(X ...

...ield of [[functional analysis]], '''DF-spaces''', also written '''(''DF'')-spaces''' are [[locally convex]] [[topological vector space]] having a property th DF-spaces were first defined by [[Alexander Grothendieck]] and studied in detail by h ...

...a property similar to the definition of [[totally bounded]] subsets. These spaces were introduced by [[Alexander Grothendieck]]. * Vector subspace of Schwartz spaces are Schwartz spaces. ...

==A sequence of elements {''a_n''} in a topological space (''Y'')== * '''[[Limit of a sequence|Convergence]]''', or "topological convergence" for emphasis (i.e. the exist ...

...in [[metric space]] that contains all [[separable space|separable]] metric spaces in a particularly nice manner. This [[mathematics]] concept is due to [[Pav ...''Urysohn universal''<ref>{{citation|title=Geometric embeddings of metric spaces|url=http://www.math.jyu.fi/research/reports/rep90.ps|author=Juha Heinonen|d ...

...following theorem: A Banach space is ''B''-convex [[if and only if]] every sequence of [[statistical independence|independent]], symmetric, uniformly bounded a ...uivalent to a number of other important properties in the theory of Banach spaces. Being '''B-convex''' and having '''Rademacher type''' <math>p>1</math> we ...

...nilpotent space. The odd-dimensional real projective spaces are nilpotent spaces, while the projective plane is not. ...n two nilpotent space is a weak homotopy equivalence. For simply connected spaces, this theorem recovers a well-known corollary to the [[Whitehead theorem|Wh ...