C space
Space of bounded sequences
In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences of real numbers or complex numbers. When equipped with the uniform norm:
the space becomes a Banach space. It is a closed linear subspace of the space of bounded sequences, , and contains as a closed subspace the Banach space of sequences converging to zero. The dual of is isometrically isomorphic to as is that of In particular, neither nor is reflexive.
In the first case, the isomorphism of with is given as follows. If then the pairing with an element in is given by
This is the Riesz representation theorem on the ordinal .
For the pairing between in and in is given by
See also
- Sequence space – Vector space of infinite sequences
References
- Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
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Banach space topics
- Asplund
- Banach
- Banach lattice
- Grothendieck
- Hilbert
- (Polynomially) Reflexive
- Riesz
- L-semi-inner product
- (B
- Strictly
- Uniformly) convex
- Uniformly smooth
- (Injective
- Projective) Tensor product (of Hilbert spaces)
- Anderson–Kadec
- Banach–Alaoglu
- Banach–Mazur
- Banach–Saks
- Banach–Schauder (open mapping)
- Banach–Steinhaus (Uniform boundedness)
- Bessel's inequality
- Cauchy–Schwarz inequality
- Closed graph
- Closed range
- Eberlein–Šmulian
- Freudenthal spectral
- Gelfand–Mazur
- Gelfand–Naimark
- Goldstine
- Hahn–Banach
- Kakutani fixed-point
- Krein–Milman
- Lomonosov's invariant subspace
- Mackey–Arens
- Mazur's lemma
- M. Riesz extension
- Parseval's identity
- Riesz's lemma
- Riesz representation
- Robinson-Ursescu
- Schauder fixed-point
- Abstract Wiener space
- Banach manifold
- Bochner space
- Convex series
- Differentiation in Fréchet spaces
- Derivatives
- Integrals
- Functional calculus
- Measures
- Weakly / Strongly measurable function
- Absolutely convex
- Absorbing
- Affine
- Balanced/Circled
- Bounded
- Convex
- Convex cone (subset)
- Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
- Linear cone (subset)
- Radial
- Radially convex/Star-shaped
- Symmetric
- Zonotope
- Absolute continuity AC
- c space
- Banach coordinate BK
- Besov
- Birnbaum–Orlicz
- Bounded variation BV
- Bs space
- Continuous C(K) with K compact Hausdorff
- Hardy Hp
- Hilbert H
- Morrey–Campanato
- ℓp
- Lp
- Schwartz
- Segal–Bargmann F
- Sequence space
- Sobolev Wk,p
- Triebel–Lizorkin
- Wiener amalgam
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