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Sequence space

In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers, typically equipped with a topology such as one induced by a norm to form a normed or Banach space.^[1]^[2] Prominent examples include the ℓᵖ spaces for 1 ≤ p ≤ ∞, where ℓ¹ comprises absolutely summable sequences with norm ‖x‖₁ = Σ |xₙ|, ℓᵖ for 1 < p < ∞ consists of sequences with finite p-norm ‖x‖ₚ = (Σ |xₙ|ᵖ)^{1/p}, and ℓ∞ denotes bounded sequences with norm ‖x‖∞ = sup |xₙ|.^[2]^[1] These spaces are complete under their respective norms, making them Banach spaces, and they arise naturally as Lᵖ spaces over the counting measure on the natural numbers.^[2] Other key sequence spaces are c, the space of convergent sequences normed by the sup norm, and its closed subspace c₀ of sequences converging to zero.^[1]^[2] Sequence spaces play a foundational role in the study of linear operators and functional analysis, serving as models for infinite-dimensional phenomena and illustrating concepts like reflexivity (for 1 < p < ∞ in ℓᵖ) and duality, where the dual of ℓ¹ is isometrically isomorphic to ℓ∞.^[2]^[1] They also illustrate the representation of continuous linear functionals, for instance, where functionals on c correspond to absolutely convergent series.^[1]

Fundamental Definitions

General concept of sequence spaces

In functional analysis, a sequence is defined as a function from the natural numbers \mathbb{N} to a field K, typically the real numbers \mathbb{R} or complex numbers \mathbb{C}.^[3] Such a sequence x = (x_n)_{n \in \mathbb{N}} assigns to each index n an element x_n \in K, forming an infinite ordered list of values from the field. This perspective views sequences as elements of the set of all functions K^{\mathbb{N}}, providing a foundational structure for studying infinite-dimensional spaces. A sequence space is a subset of K^{\mathbb{N}} that constitutes a vector space over K under the operations of pointwise addition and scalar multiplication. For sequences x = (x_n) and y = (y_n) in the space, and scalar \alpha \in K, the addition is defined by (x + y)_n = x_n + y_n for all n \in \mathbb{N}, and scalar multiplication by (\alpha x)_n = \alpha x_n for all n \in \mathbb{N}. These operations ensure closure, associativity, commutativity, and the existence of zero and additive inverses within the subset, satisfying the vector space axioms.^[4] The concept of sequence spaces originated in early 20th-century functional analysis, with key developments by Maurice Fréchet in his 1906 thesis, where he introduced metric structures on spaces of sequences, and later formalized by Stefan Banach in the 1920s through his work on complete normed linear spaces.^[5]

Space of all real or complex sequences

The space of all sequences of real numbers, denoted \mathbb{R}^\mathbb{N} or \omega, consists of all functions from the natural numbers \mathbb{N} to \mathbb{R}, and analogously \mathbb{C}^\mathbb{N} or \omega for complex numbers.^[6]^[7] This space serves as the universal sequence space, encompassing every possible infinite sequence without restrictions on growth or summability. Algebraically, \mathbb{R}^\mathbb{N} forms an infinite-dimensional vector space over \mathbb{R}, with addition and scalar multiplication defined componentwise: for sequences x = (x_n) and y = (y_n), and scalar \alpha \in \mathbb{R}, the operations are (x + y)_n = x_n + y_n and (\alpha x)_n = \alpha x_n. The standard unit sequences e_k, defined by (e_k)_n = \delta_{kn} where \delta_{kn} is the Kronecker delta (equal to 1 if k = n and 0 otherwise), form a linearly independent set that algebraically spans the proper subspace of sequences with finite support.^[8] A Hamel basis for the full space \mathbb{R}^\mathbb{N} has cardinality equal to the continuum, $2^{\aleph_0}, reflecting the uncountable dimension required to span all sequences algebraically.^[9] Topologically, the space is often equipped with the product topology, generated by the coordinate projections \pi_k: \mathbb{R}^\mathbb{N} \to \mathbb{R} given by \pi_k(x) = x_k, which makes all projections continuous. This topology is metrizable via a metric such as d(x, y) = \sum_{k=1}^\infty 2^{-k} \frac{|x_k - y_k|}{1 + |x_k - y_k|}, and complete under this metric. However, the product topology is not normable, as the space lacks a bounded neighborhood of the origin; any neighborhood contains sets unbounded in infinitely many coordinate directions. Consequently, no norm exists on [\mathbb{R}^\mathbb{N}](/page/sequence_space) under which all coordinate projections are continuous, preventing the space from being structured as a Banach space and leading to pathological features such as the ubiquity of discontinuous linear functionals (constructed via the Hamel basis using the axiom of choice).^[10]^[11] As the ambient space, \mathbb{R}^\mathbb{N} contains all restricted sequence spaces, such as the \ell_p spaces for $1 \leq p \leq \infty, as vector subspaces.

Normed Sequence Spaces

ℓ_p spaces for 1 ≤ p < ∞

The \ell_p spaces for $1 \leq p < \infty consist of all sequences x = (x_n)_{n=1}^\infty of complex (or real) numbers such that \sum_{n=1}^\infty |x_n|^p < \infty. These form a vector space under componentwise addition and scalar multiplication, equipped with the p-norm defined by

\|x\|_p = \left( \sum_{n=1}^\infty |x_n|^p \right)^{1/p}.

^[12] This norm satisfies the usual properties, including the triangle inequality, making (\ell_p, \|\cdot\|_p) a normed linear space.^[12] For p=1, \ell_1 is the space of absolutely summable sequences, where \|x\|_1 = \sum_{n=1}^\infty |x_n| represents the total variation of the sequence.^[12] For p=2, \ell_2 comprises square-summable sequences, with the norm \|x\|_2 = \sqrt{\sum_{n=1}^\infty |x_n|^2} corresponding to the Euclidean length in infinite dimensions.^[12] In general, as p increases toward \infty, the condition \sum |x_n|^p < \infty becomes less restrictive on the decay of |x_n|, allowing sequences with slower convergence to remain in the space.^[12] Each \ell_p is complete with respect to the p-norm, meaning every Cauchy sequence converges to an element in \ell_p; this follows from the convergence of series in the p-norm, establishing \ell_p as a Banach space.^[12] The proof relies on showing that partial sums or limits of Cauchy sequences remain p-summable, using the dominated convergence theorem or direct estimates on tails.^[12] For $1 < p < \infty, the unit ball \{x \in \ell_p : \|x\|_p \leq 1\} is strictly convex: if \|x\|_p = \|y\|_p = 1 and x \neq y, then \|(x + y)/2\|_p < 1.^[12] This property arises from the strict convexity of the function t \mapsto |t|^p on \mathbb{R}, ensuring no line segments lie on the boundary of the unit ball except at endpoints.^[12] In contrast, \ell_1 lacks strict convexity, as its unit ball contains line segments, such as between (1,0,0,\dots) and (0,1,0,\dots).^[12]

ℓ_∞ and bounded sequences

The space \ell_\infty is defined as the set of all bounded sequences x = (x_n)_{n=1}^\infty of complex or real numbers, that is, sequences for which \sup_{n \in \mathbb{N}} |x_n| < \infty, equipped with the supremum norm \|x\|_\infty = \sup_{n \in \mathbb{N}} |x_n|.^[13] This norm satisfies the properties of a norm, making \ell_\infty a normed vector space, and it induces a metric under which the space is complete, thus establishing \ell_\infty as a Banach space.^[14] Unlike the \ell_p spaces for $1 \leq p < \infty, which are separable, \ell_\infty is not separable; any dense subset must be uncountable, reflecting its larger structure.^[15] The Hamel basis (algebraic basis) of \ell_\infty has cardinality equal to the continuum $2^{\aleph_0}, consistent with the dimension of infinite-dimensional Banach spaces of this cardinality.^[16] The closed unit ball \{x \in \ell_\infty : \|x\|_\infty \leq 1\} is not strictly convex, as it contains line segments on its boundary; for instance, the convex combination of the constant sequence x_n = 1 for all n and the alternating sequence y_n = (-1)^n lies entirely on the unit sphere.^[17] For each $1 \leq p < \infty, the space \ell_p is a proper subspace of \ell_\infty, since every p-summable sequence is bounded (as terms tend to zero), yielding a continuous inclusion \ell_p \hookrightarrow \ell_\infty.^[13] However, this inclusion is not dense in the norm topology of \ell_\infty, as the finite support sequences c_{00} (dense in \ell_p) fail to approximate arbitrary bounded sequences uniformly; density of such subspaces holds only in weaker topologies, such as the weak or pointwise convergence topologies. Subspaces like the convergent sequences c form closed proper subsets of \ell_\infty.

Specialized Sequence Spaces

Spaces of convergent, null, and finitely supported sequences

The space c consists of all convergent sequences of complex numbers (or real numbers), equipped with the supremum norm \|x\|_\infty = \sup_{n \in \mathbb{N}} |x_n|. This norm makes c a Banach space, as it is a closed subspace of the space \ell_\infty of all bounded sequences under the same norm.^[18]^[19] A sequence (x^{(k)})_{k \in \mathbb{N}} in c converges to x \in c in this norm if and only if it converges uniformly to x, meaning \sup_n |x^{(k)}_n - x_n| \to 0 as k \to \infty. Since every element of c is bounded, the space embeds isometrically into \ell_\infty, and the closedness follows from the fact that if a sequence in c converges in \ell_\infty, the limit must also converge (to the common limit of the coordinates).^[20] The subspace c_0 of c comprises those sequences in c that converge to zero, i.e., \lim_{n \to \infty} x_n = 0. It inherits the supremum norm from \ell_\infty and is likewise a closed subspace, hence a Banach space.^[20]^[21] For example, the standard basis vectors truncated appropriately lie in c_0, but sequences like the constant 1 do not. Convergence in c_0 under the sup norm requires uniform convergence to the zero sequence, ensuring the tails vanish uniformly. The inclusion c_0 \subset c \subset \ell_\infty is proper, with c_0 distinguished by its elements having limit zero.^[22] The space c_{00} (also denoted \phi) consists of all sequences with only finitely many nonzero terms, forming a subspace of both c_0 and c. It is equipped with the sup norm but is not complete under this norm, as it is not closed in \ell_\infty. However, c_{00} is dense in c_0 with respect to the sup norm: for any x \in c_0, one can approximate x by truncating its tail beyond some large N, with the approximation error bounded by \sup_{n > N} |x_n|, which tends to zero as N \to \infty. Similarly, c_{00} is dense in \ell_p for $1 \leq p < \infty under the respective \ell_p norms, serving as a fundamental dense algebraic subspace in these settings.^[20]^[21]^[22] On c, c_0, and c_{00}, the topology induced by the sup norm corresponds to uniform convergence of sequences, which strengthens the pointwise (coordinatewise) convergence topology inherited from the product space \mathbb{C}^\mathbb{N}. While pointwise convergence defines a coarser topology where a net converges if it does so at each coordinate, the sup norm ensures uniform bounds on the convergence rate across all coordinates, making it Hausdorff and metrizable on these subspaces.^[20]^[21]

Finite-dimensional sequence spaces

Finite-dimensional sequence spaces, often denoted as \ell_p^n for $1 \leq p \leq \infty and n \in \mathbb{N}, consist of all sequences (x_1, \dots, x_n) with entries in \mathbb{R} or \mathbb{C}, equipped with the \ell_p norm defined by

\|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right)^{1/p}

for $1 \leq p < \infty, and \|x\|_\infty = \max_{1 \leq i \leq n} |x_i| for p = \infty.^[23] These spaces are isomorphic to the standard Euclidean space \mathbb{R}^n or \mathbb{C}^n as vector spaces, where the isomorphism identifies sequences of fixed length n with n-tuples.^[24] Each \ell_p^n embeds naturally into the space c_{00} of finitely supported sequences by extending the sequence with zeros beyond the nth position, preserving the vector space structure and the norm since only finitely many terms are nonzero.^[24] The union over all n \in \mathbb{N} of these embedded \ell_p^n spans c_{00}, which is the algebraic direct sum of the one-dimensional spaces generated by the standard basis vectors e_k = (0, \dots, 0, 1, 0, \dots) with 1 in the kth position.^[24] As finite-dimensional normed spaces, each \ell_p^n is a Banach space, being complete with respect to the \ell_p norm.^[25] A key property of these spaces is that all norms on \ell_p^n—including the \ell_q norms for different q—are equivalent, meaning for any $1 \leq p, q \leq \infty, there exist positive constants c, C (depending on n, p, q) such that c \|x\|_p \leq \|x\|_q \leq C \|x\|_p for all x \in \ell_p^n.^[25] This equivalence arises because any two norms on a finite-dimensional vector space induce the same topology and are related by bounded linear operators with respect to a fixed basis, such as the standard basis.^[25] These finite-dimensional spaces serve as foundational approximations in functional analysis, where subspaces like \ell_p^n (embedded in larger sequence spaces) enable numerical methods to approximate solutions to problems in infinite-dimensional settings, such as solving linear systems or optimizing in \ell_p or c_0.^[24]

Inclusion and Embedding Properties

Monotonicity of ℓ_p spaces with respect to p

In the theory of sequence spaces, the ℓ_p spaces exhibit a monotonicity property with respect to the parameter p. Specifically, for 1 ≤ p < q ≤ ∞, the space ℓ_p is continuously embedded into ℓ_q, meaning ℓ_p ⊂ ℓ_q as sets, and the inclusion map is a bounded linear operator with ||x||_q ≤ ||x||_p for all x ∈ ℓ_p.^[12] This embedding reflects the fact that sequences with finite p-norm for smaller p necessarily have finite q-norm for larger q, due to the behavior of the p-norms on sequences where terms decay to zero. To see the norm inequality, assume ||x||p = 1. Then ||x||∞ = \sup |x_j| ≤ 1, as otherwise ||x||_p ≥ \sup |x_j| > 1. Since q > p and 0 ≤ |x_j| ≤ 1, |x_j|^q ≤ |x_j|^p for each j. Thus, ||x||_q^q = \sum |x_j|^q ≤ \sum |x_j|^p = 1, so ||x||_q ≤ 1 = ||x||_p. For q = ∞, the bound is direct from \sup |x_j| ≤ 1.^[12] The inclusions are strict, meaning ℓ_p \subsetneq ℓ_q. A representative example is the harmonic sequence x = (1/n)_{n=1}^∞, which belongs to ℓ_q \ ℓ_p whenever p ≤ 1 < q, as \sum (1/n)^q < ∞ for q > 1 but \sum (1/n)^p = ∞ for p ≤ 1.^[26] For general 1 < p < q ≤ ∞, similar constructions yield sequences in ℓ_q whose p-norms diverge, such as appropriately scaled versions with decay rate between 1/q and 1/p, confirming the proper containment.^[26] In the boundary case as p → ∞, the spaces ℓ_p increase monotonically and approach ℓ_∞ in the sense of inclusion: every element of ℓ_p lies in ℓ_∞, and the union \bigcup_{1 ≤ p < ∞} ℓ_p is dense in c_0 with respect to the sup norm, though ℓ_∞ properly contains sequences not in any ℓ_p (e.g., the constant sequence (1,1,...)).^[12]^[27] This limiting behavior underscores the transition from summability conditions to mere boundedness.

Dense subspaces and completions

In the normed sequence spaces \ell_p for $1 \leq p < \infty, the subspace c_{00} consisting of all finitely supported sequences is dense with respect to the \ell_p-norm.^[28] More specifically, the countable set of finitely supported sequences with rational entries forms a dense subset of \ell_p, establishing the separability of these spaces.^[29] This countable dense subset arises because the rational numbers are dense in the reals, allowing approximation of any sequence in \ell_p by truncating tails and rationalizing coordinates while controlling the norm.^[29] Similarly, in the space c_0 of sequences converging to zero equipped with the supremum norm, the subspace c_{00} is dense.^[28] Any sequence in c_0 can be approximated by its finite initial segments, which belong to c_{00}, since the tail vanishes uniformly.^[29] In contrast, \ell_\infty, the space of bounded sequences with the supremum norm, admits no such countable dense subset and is non-separable.^[29] For the spaces \ell_p with $0 < p < 1, the \ell_p-quasi-norm \|x\|_p = \left( \sum_{n=1}^\infty |x_n|^p \right)^{1/p} fails to satisfy the triangle inequality but induces a complete quasi-metric d(x,y) = \|x - y\|_p^p.^[30] Thus, \ell_p is a complete quasi-Banach space under this structure, with completeness verified by showing that every quasi-Cauchy sequence converges.^[30] In general, the completion of any normed or quasi-normed sequence space is obtained by adjoining limits of Cauchy (or quasi-Cauchy) sequences, forming equivalence classes under the relation that two sequences converge to the same limit if their difference tends to zero in norm.^[29] These dense subspaces, such as c_{00}, mirror the role of step functions in the analogous L_p function spaces, where finite combinations provide approximations that establish density and separability for $1 \leq p < \infty.^[31]

Hilbert and Banach Space Structures

ℓ_2 as a Hilbert space

The space \ell_2 consists of all complex sequences x = (x_n)_{n=1}^\infty such that \sum_{n=1}^\infty |x_n|^2 < \infty, equipped with the inner product \langle x, y \rangle = \sum_{n=1}^\infty x_n \overline{y_n}, where the series converges absolutely by the Cauchy-Schwarz inequality.^[32] This inner product induces the norm \|x\|_2 = \sqrt{\langle x, x \rangle} = \left( \sum_{n=1}^\infty |x_n|^2 \right)^{1/2}, making \ell_2 a complete inner product space, hence a Hilbert space.^[32] As the prototypical separable infinite-dimensional Hilbert space, \ell_2 serves as a fundamental model for studying inner product structures in infinite dimensions. The standard basis \{e_n\}_{n=1}^\infty, where e_n has a 1 in the n-th position and 0 elsewhere, forms a countable orthonormal set in [\ell_2](/page/l2-space) since \langle e_m, e_n \rangle = \delta_{mn}. This basis is complete, meaning its linear span is dense in [\ell_2](/page/l2-space), which follows from the fact that any x \in \ell_2 can be approximated by finite partial sums \sum_{k=1}^N x_k e_k. Parseval's identity holds for this basis: for any x \in \ell_2, \|x\|_2^2 = \sum_{n=1}^\infty |\langle x, e_n \rangle|^2 = \sum_{n=1}^\infty |x_n|^2, reflecting the preservation of norms under orthonormal expansions. Every separable infinite-dimensional Hilbert space is unitarily isomorphic to \ell_2, meaning there exists a unitary operator U (preserving the inner product) that maps an orthonormal basis of the space onto \{e_n\}.^[33] This isomorphism underscores \ell_2's universality among such spaces.^[33] Additionally, the Riesz representation theorem characterizes the dual space of \ell_2: every continuous linear functional f: \ell_2 \to \mathbb{C} is of the form f(x) = \langle x, y \rangle for a unique y \in \ell_2, with \|f\| = \|y\|_2.^[34]

Banach space properties of ℓ_1 and other ℓ_p spaces

The space \ell_1 exhibits the Schur property, which states that every sequence in \ell_1 that converges weakly also converges in the \ell_1-norm. This property distinguishes \ell_1 from most other Banach spaces. Additionally, the extreme points of the closed unit ball in \ell_1 are the sequences with a single non-zero entry of modulus 1 and zeros elsewhere; any other point in the unit ball can be expressed as a nontrivial convex combination of such points.^[35] In contrast, for $1 < p < \infty, the spaces \ell_p are reflexive Banach spaces, meaning their canonical embedding into their bidual is surjective. These spaces are also uniformly convex, a property that implies strict convexity of the unit ball and ensures that the norm is Gateaux differentiable almost everywhere. Consequently, \ell_p for $1 < p < \infty possess the Radon-Nikodym property, allowing the representation of certain vector measures by Bochner integrable functions. The space \ell_\infty lacks reflexivity, as its bidual properly contains it, and it fails the Schur property, since sequences like the standard basis vectors converge weakly to zero but not in norm. Unlike \ell_\infty, \ell_1 forms a Banach algebra under the convolution product defined by (a * b)_n = \sum_{k=0}^n a_k b_{n-k}, with the \ell_1-norm satisfying \|a * b\|_1 \leq \|a\|_1 \|b\|_1, ensuring uniform boundedness of the algebra multiplication.

Advanced Topics

Dual spaces of sequence spaces

The dual space of the sequence space \ell_1, equipped with the \ell_1-norm, is isometrically isomorphic to \ell_\infty. Every continuous linear functional \phi on \ell_1 can be represented uniquely as \phi(x) = \sum_{n=1}^\infty x_n y_n for some y = (y_n) \in \ell_\infty, where the duality pairing satisfies \|\phi\| = \|y\|_\infty.^[36] For $1 < p < \infty, the dual space of \ell_p is \ell_q, where \frac{1}{p} + \frac{1}{q} = 1. The isomorphism is given by the pairing \langle x, y \rangle = \sum_{n=1}^\infty x_n y_n for x \in \ell_p and y \in \ell_q, and it preserves the norms: \|\phi_y\| = \|y\|_q. This representation holds isometrically, establishing a natural duality between these spaces.^[36]^[37] The dual of c_0, the space of sequences converging to zero under the supremum norm, is isometrically isomorphic to \ell_1. Continuous linear functionals on c_0 take the form \phi(x) = \sum_{n=1}^\infty x_n a_n for a = (a_n) \in \ell_1, with the operator norm equaling \|a\|_1.^[36]^[37] In contrast, the dual of c, the space of all convergent sequences with the supremum norm, is more complex than a simple \ell_q space and involves additional structure akin to measures on the natural numbers. Specifically, it can be identified with \ell_1 \oplus \mathbb{R} under an appropriate norm, where functionals are of the form \phi(x) = \sum_{n=1}^\infty x_n y_n + \alpha \lim_{n \to \infty} x_n for y \in \ell_1 and \alpha \in \mathbb{R}, though this space is isometrically isomorphic to \ell_1. The norm is \| (y, \alpha) \| = \|y\|_1 + |\alpha|.^[38] Regarding reflexivity, the spaces \ell_p are reflexive for $1 < p < \infty, meaning \ell_p^{**} \cong \ell_p isometrically, due to the uniform convexity of their unit balls. However, \ell_1 and \ell_\infty are not reflexive, as their biduals properly contain them: (\ell_1)^* = \ell_\infty but (\ell_\infty)^* is larger, isomorphic to the space of bounded finitely additive measures on \mathbb{N}.^[39] Similarly, neither c_0 nor c is reflexive.^[40]

Other notable sequence spaces

Weighted \ell_p spaces generalize the classical \ell_p spaces by incorporating positive weights w_n > 0, defined as the set of sequences x = (x_n) such that \sum_{n=1}^\infty |x_n w_n|^p < \infty for $1 \leq p < \infty, equipped with the norm \|x\|_{p,w} = \left( \sum_{n=1}^\infty |x_n w_n|^p \right)^{1/p}.^[41] These spaces form Banach spaces and reduce to the standard \ell_p when w_n = 1 for all n. A common example uses weights w_n = (n+1)^\alpha for \alpha \in \mathbb{R}, yielding \ell_{p,\alpha} with norm \|x\|_{p,\alpha} = \left( \sum_{k=0}^\infty |x_k|^p (k+1)^\alpha \right)^{1/p} < \infty, which models sequences corresponding to analytic functions in the unit disk.^[41] In operator theory, these spaces facilitate the study of bounded operators like shifts and multipliers on Hardy spaces H_p.^[41] Orlicz sequence spaces extend \ell_p spaces using a convex function \Phi: [0,\infty) \to [0,\infty) with \Phi(0) = 0 and \Phi(x) \to \infty as x \to \infty, known as an Orlicz function. The space \ell^\Phi consists of sequences x such that \sum_{n=1}^\infty \Phi(|x_n|/c) < \infty for some c > 0, with the Luxemburg norm \|x\|_\Phi = \inf \{ c > 0 : \sum_{n=1}^\infty \Phi(|x_n|/c) \leq 1 \}.^[42] This generalizes \ell_p by taking \Phi(x) = x^p/p, and the framework relies on the convexity of \Phi to ensure the triangle inequality.^[42] An example is the exponential class with \Phi_\alpha(x) = \exp(x^\alpha) - 1 for $0 < \alpha \leq 2, which captures sub-Gaussian tails in probability theory, bounding maxima of random variables in empirical processes.^[42] Introduced by Orlicz in the 1930s, these spaces apply to moment conditions in probability distributions beyond power moments.^[42] Köthe sequence spaces are perfect sequence spaces forming Banach lattices, defined via a matrix P = (p_{mn}) of non-negative entries where the space \lambda(P) includes sequences x = (x_n) satisfying \sup_m \sum_n |x_n| p_{mn} < \infty for each fixed m, with the norm \|x\| = \sup_m \sum_n |x_n| p_{mn}.^[43] The matrix P determines inclusions through lattice properties: if x, y \in \lambda(P), then the sequences with coordinates \min(|x_n|, |y_n|) and \max(|x_n|, |y_n|) also belong to \lambda(P), ensuring solidity.^[43] The dual space \lambda(P)^\sim is given by the "inverse" matrix in a Köthe-Toeplitz sense.^[43] Developed by Köthe in the mid-20th century, these spaces model test functions and distributions in operator theory, serving as ideals in the algebra of bounded operators on \ell^\infty and in nuclear space constructions.^[43]

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[PDF] What Is Uniform Convexity? - OSU Math
Jun 6, 2024 · The ideology here is that for convex optimization purposes, strict convexity gives unique- ness, and uniform convexity gives existence (and also ...Missing: l_p infinity
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[PDF] Real Analysis - Harvard Mathematics Department
We also have c = c(N) = C(N∪∞), the space of convergent sequences with the sup norm, and c0 ⊂ c, the space of sequences converging to zero. The Lp spaces ...Missing: l_infty | Show results with:l_infty
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[PDF] SAMSA Masamu Program - Auburn University
Let c be the space of convergent sequences and l∞ the space of bounded sequences, both with the supremun norm i.e. (c, k.k∞) and (l∞, k.k∞). We are ...Missing: l_infty | Show results with:l_infty
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[PDF] Functional Analysis, Math 7320 Lecture Notes from October 04, 2016
Oct 4, 2016 · The subspace c and c0 are closed subspaces of l∞ (and hence are Banach spaces). The space c00 is only a subspace in c0, but not closed in ...Missing: ell_infty | Show results with:ell_infty
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[PDF] MAA 6617 Course Notes Spring 2025 1. Normed vector spaces
Jan 17, 2025 · and this quantity is a norm making ℓp into a Banach space. When p = ∞, we define L∞(µ) to be the set of all functions f : X → K with the.
[23]
[PDF] short review of metrics, norms, and convergence - Christopher Heil
The vectors in c00 are sometimes called finite sequences because they contain at most finitely many nonzero components. If we fix a particular value of p, then ...Missing: ell_p | Show results with:ell_p
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### Summary of Finite-Dimensional ℓ_p Spaces from the Document
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[PDF] Lecture 3 - Functional analysis
basis for any of the spaces c0 or `p (1 ≤ p ≤ ∞), because it spans only the space c00 of finitely supported sequences, i.e.. • c00 = {(xn)ю n=1 ∈ KN : xn ...
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[PDF] EQUIVALENCE OF NORMS 1. Introduction Let K be a field and
All norms on a finite-dimensional vector space over a complete valued field are equivalent. Proof. Let (K,|·|) be a complete valued field and V be a K-vector ...
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Summary of each segment:
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### Summary of Dense Subspaces in Sequence Spaces
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[PDF] FUNCTIONAL ANALYSIS - ETH Zürich
Jun 8, 2017 · These are notes for the lecture course “Functional Analysis I” held by the second author at ETH Zürich in the fall semester 2015.Missing: ℓ_p ℓ_q
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Quasi-Banach space for the sequence space lp where 0<p<1
The aim of this paper is to introduce the concept of a quasi-Banach space for the sequence space , p 0 p 1 . This concept is based on the important ...Missing: ell_p | Show results with:ell_p
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[PDF] Lp-SPACES FOR 0 <p< 1 - KEITH CONRAD
A Banach space is a vector space V equipped with a norm || · || such that, with respect to the metric defined by d(v, w) = ||v − w||, V is complete. Example 2.5 ...
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MATHEMATICA tutorial. Part 4.2: Hilbert spaces
Sep 26, 2025 · The set of such sequences is denoted by ℓ2 or ℓ². More precisely ... It turns out that the ℓ2 or ℓ2 norm is generated by the inner product:.<|control11|><|separator|>
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[PDF] Hilbert spaces
A maximal orthonormal sequence in a separable Hilbert space is called a complete orthonormal basis. This notion of basis is not quite the same as in the finite ...
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[PDF] hilbert spaces and the riesz representation theorem - UChicago Math
The Riesz representation theorem is a powerful result in the theory of Hilbert spaces which classifies continuous linear functionals in terms of the inner ...
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[PDF] Chapter 6. Duality - 6.1. Examples of Dual Spaces
May 20, 2015 · (a) The dual of c0 (the space of all sequences which converge to 0, with the sup ... By Example 2.18, c00 is a dense subset of c0. Since φ is ...<|separator|>
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Below is a merged summary of the dual spaces of sequence spaces based on the provided segments from the document at https://uomustansiriyah.edu.iq/media/lectures/9/9_2017_09_30!12_00_39_PM.pdf. Since the document primarily focuses on topics like compact operators, Hilbert spaces, Banach algebras, C*-algebras, spectral theory, and Fourier transforms, with limited direct content on dual spaces of sequence spaces (e.g., $ l_p $, $ c_0 $, $ c $), I will consolidate all relevant information into a comprehensive response. Where specific details are missing or inconsistent across segments, I will note the gaps and provide the most detailed synthesis possible. To maximize density and clarity, I will use tables in CSV format where appropriate, followed by a narrative summary.
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[PDF] PROBLEMS ON c AND c Definition 1. Sequences with unspecified ...
(c) Show that c is not isometrically isomorphic to c0 (even so, note that their dual spaces c∗ and c0. ∗ are each isometrically isomorphic to ℓ1 by Problem ...
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[PDF] On Weighted Sequence Spaces - ODU Digital Commons
Oct 2, 2024 · The main theorem of this chapter gives exact conditions on the parameters for the inclusion relation ℓp,α ⊆ ℓr,β to hold. Its proof will be ...Missing: l_p | Show results with:l_p
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[PDF] Orlicz spaces - Yale Statistics and Data Science
Jul 6, 2025 · Section 5.2 describes some useful properties of orlicz functions Ψ on R+, the convex functions that are used to define the LΨ spaces. They are.
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https://math.hse.ru/data/2017/11/20/1161024002/fa_list_03.pdf