ℓ^∞

ℓ^∞
Name	ℓ^∞
Caption	Space of bounded sequences
Type	Banach space
Field	Functional analysis
Elements	Bounded real or complex sequences
Operations	Pointwise addition and scalar multiplication
Norm	Supremum norm

ℓ^∞

ℓ^∞ is the Banach space of all bounded scalar sequences equipped with the supremum norm. It arises naturally in Functional analysis, Measure theory, Harmonic analysis, Operator theory, and Probability theory as a fundamental example of a nonseparable, commutative C*-algebra analogue and a prototypical dual object. Its structure connects to classical results by Steinhaus, Banach, Hahn–Banach theorem, Grothendieck, and Schauder.

Definition and basic properties

Elements of ℓ^∞ are sequences (x_n) with x_n in either ℝ or ℂ such that sup_n |x_n| < ∞; pointwise addition and scalar multiplication make it a vector space. The supremum norm ||x||_∞ = sup_n |x_n| induces a metric that renders ℓ^∞ complete, yielding a Banach space studied alongside L^p spaces, c_0, ℓ^1, ℓ^2, and C(K) spaces. As a concrete example, constant sequences, eventually zero sequences, and bounded oscillatory sequences belong to ℓ^∞; classical sequences like those used by Dirichlet, Cauchy, and Möbius appear as elements when bounded. ℓ^∞ contains copies of finite-dimensional spaces such as ℝ^n and ℂ^n.

Normed space and Banach space structure

With the supremum norm, ℓ^∞ is a normed space whose completeness is routinely established via Cauchy sequence arguments found in expositions by Banach, Riesz, Fréchet, and Dieudonné. ℓ^∞ is nonseparable: the uncountable family of sequences taking values 0 or 1 at each index provides a standard argument attributed to Kuratowski and Ulam. As a Banach space, ℓ^∞ plays a central role in counterexamples such as the failure of the separable dual property encountered in work by Enflo and in constructions related to Johnson and Lindenstrauss.

Dual space and bounded linear functionals

The continuous dual (ℓ^∞)* strictly contains ℓ^1 via the canonical pairing summation ∑ x_n y_n for y in ℓ^1, a fact articulated in the developments by Hahn–Banach theorem, Banach–Alaoglu theorem, and Goldstine. The full dual is larger and can be described using finitely additive measures or via the space ba of bounded finitely additive set functions on ℕ studied by Yosida and Hewitt. The existence of nonprincipal ultrafilters on ℕ, connected to results by Tarski and Ultrafilter lemma, leads to Banach limits, which are translation-invariant positive functionals on ℓ^∞ constructed using tools related to Axiom of Choice and Hahn–Banach theorem. These phenomena are linked to the study of weak* topology and examples by Sakai and Kadison.

Subspaces and canonical examples

Important closed subspaces include c_0 (sequences converging to zero), which is the closed ideal generated by finitely supported sequences and appears in the literature from Schauder and Pelczynski. ℓ^∞/c_0 yields the Banach space known as the Calkin-like quotient or the Banach space of bounded sequences modulo null sequences, related to the Stone–Čech remainder βℕ\ℕ via work by Stone and Čech. The algebra ℓ^∞ with pointwise multiplication is a commutative C*-algebra isomorphic to C(βℕ) per the Gelfand–Naimark framework employed by Gelfand and Naimark. Finite-dimensional subspaces, complemented copies, and embeddings studied by Bessaga and Pełczyński provide canonical examples and counterexamples in Banach space theory.

Topological and geometric properties

Topologically, the unit ball of ℓ^∞ is compact in the weak* topology by the Banach–Alaoglu theorem when ℓ^∞ is viewed as the dual of ℓ^1, but not compact in the norm topology, a contrast emphasized in treatments by Kelley and Dunford. Geometrically, ℓ^∞ lacks properties like uniform convexity and rotundity examined by Milman and Kadec, and its extreme points and faces have been characterized in the tradition of Krein–Milman theorem analyses. The space admits many projections and is rich in complemented and uncomplemented subspaces, topics studied by Sobczyk and Pelczynski.

Applications and connections to functional analysis

ℓ^∞ appears in the theory of bounded linear operators, spectral theory for multiplication operators, and C*-algebra representations featuring in works by Gelfand, Sakai, Kadison, and Dixmier. In ergodic theory and topological dynamics, bounded sequences model observables and link to recurrence results by Poincaré and Birkhoff; Banach limits on ℓ^∞ yield invariant means related to amenability studied by von Neumann and Day. In probability and stochastic processes, ℓ^∞ serves as a sample space for bounded sample paths in formulations by Kolmogorov and Doob. Connections to modern research include isomorphic classification problems addressed by Johnson, Lindenstrauss, Maurey', and applications in signal processing and numerical analysis referenced in work by Courant and Krylov.

Category:Banach spaces