{"history":[],"article":{"id":"e4f664a1-dac2-48d4-a44d-9fd954b64035","title":"Schauder basis","slug":"schauder-basis","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **Schauder [basis](https://bliptext.com/articles/basis-linear-algebra)** or **countable [basis](https://bliptext.com/articles/basis-linear-algebra)** is similar to the usual ([Hamel](https://bliptext.com/articles/hamel-[basis](https://bliptext.com/articles/basis-linear-algebra))) [basis](https://bliptext.com/articles/basis-linear-algebra) of a [vector space](https://bliptext.com/articles/vector-space); the difference is that [Hamel](https://bliptext.com/articles/hamel-[basis](https://bliptext.com/articles/basis-linear-algebra)) bases use [linear combinations](https://bliptext.com/articles/linear-combination) that are finite sums, while for Schauder bases they may be infinite sums.\nThis makes Schauder bases more suitable for the analysis of infinite-dimensional [topological vector spaces](https://bliptext.com/articles/topological-vector-space) including [Banach spaces](https://bliptext.com/articles/banach-space).\nSchauder bases were described by [Juliusz Schauder](https://bliptext.com/articles/juliusz-schauder) in 1927, although such bases were discussed earlier.\nFor example, the [Haar basis](https://bliptext.com/articles/haar-basis) was given in 1909, and [Georg Faber](https://bliptext.com/articles/georg-faber) discussed in 1910 a basis for [continuous functions](https://bliptext.com/articles/continuous-function) on an [interval](https://bliptext.com/articles/interval-mathematics), sometimes called a **Faber–Schauder system**.\n\n# Definitions\nLet V denote a [topological vector space](https://bliptext.com/articles/topological-vector-space) over the [field](https://bliptext.com/articles/field-mathematics) F.\nA **Schauder basis** is a [sequence](https://bliptext.com/articles/sequence) {bn} of elements of V such that for every element v ∈ V there exists a unique [sequence](https://bliptext.com/articles/sequence) {αn} of scalars in F so that The convergence of the infinite sum is implicitly that of the ambient topology, i.e., but can be reduced to only [weak convergence](https://bliptext.com/articles/weak-topology) in a [normed vector space](https://bliptext.com/articles/normed-vector-space) (such as a [Banach space](https://bliptext.com/articles/banach-space)).\nUnlike a [Hamel basis](https://bliptext.com/articles/hamel-basis), the elements of the basis must be ordered, since the series may not converge [unconditionally](https://bliptext.com/articles/unconditional-convergence).\nNote that some authors define Schauder bases to be countable (as above), while others use the term to include uncountable bases.\nIn either case, the sums themselves always are countable.\nAn uncountable Schauder basis is a [linearly ordered set](https://bliptext.com/articles/linearly-ordered-set) rather than a sequence, and each sum inherits the order of its terms from this linear ordering.\nThey can and do arise in practice.\nAs an example, a [separable](https://bliptext.com/articles/separable-space) [Hilbert space](https://bliptext.com/articles/hilbert-space) can only have a countable Schauder basis, but a non-[separable](https://bliptext.com/articles/separable-space) [Hilbert space](https://bliptext.com/articles/hilbert-space) may have an uncountable one.\nThough the definition above technically does not require a normed space, a norm is necessary to say almost anything useful about Schauder bases.\nThe results below assume the existence of a norm.\nA Schauder basis {bn}n ≥ 0 is said to be **normalized** when all the basis vectors have norm 1 in the Banach space V.\nA sequence {xn}n ≥ 0 in V is a **basic sequence** if it is a Schauder basis of its [closed linear span](https://bliptext.com/articles/linear-span).\nTwo Schauder bases, {bn} in V and {cn} in W, are said to be **equivalent** if there exist two constants c > 0 and C such that for every [natural number](https://bliptext.com/articles/natural-number) N ≥ 0 and all sequences {αn} of scalars,\nA family of vectors in V is **total** if its [linear span](https://bliptext.com/articles/linear-span) (the [set](https://bliptext.com/articles/set-mathematics) of finite linear combinations) is [dense](https://bliptext.com/articles/dense-subset) in V.\nIf V is a [Hilbert space](https://bliptext.com/articles/hilbert-space), an **[orthogonal basis](https://bliptext.com/articles/orthogonal-basis)** is a total [subset](https://bliptext.com/articles/subset) B of V such that elements in B are nonzero and pairwise orthogonal.\nFurther, when each element in B has norm 1, then B is an **[orthonormal basis](https://bliptext.com/articles/orthonormal-basis)** of V.\n\n# Properties\nLet {bn} be a Schauder basis of a Banach space V over **F** = **R** or **C**.\nIt is a subtle consequence of the [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-functional-analysis) that the linear mappings {Pn} defined by\nare uniformly bounded by some constant C.\nWhen, the basis is called a **monotone** basis.\nThe maps {Pn} are the **[basis projections](https://bliptext.com/articles/projection-linear-algebra)**.\nLet {b*n} denote the **coordinate functionals**, where b*n assigns to every vector v in V the coordinate αn of v in the above expansion.\nEach b*n is a bounded linear functional on V.\nIndeed, for every vector v in V,\nThese functionals {b*n} are called **biorthogonal functionals** associated to the basis {bn}.\nWhen the basis {bn} is normalized, the coordinate functionals {b*n} have norm ≤ 2C in the [continuous dual](https://bliptext.com/articles/dual-space) V ′ of V.\nA Banach space with a Schauder basis is necessarily [separable](https://bliptext.com/articles/separable-space), but the converse is false.\nSince every vector v in a Banach space V with a Schauder basis is the limit of Pn(v), with Pn of finite rank and uniformly bounded, such a space V satisfies the [bounded approximation property](https://bliptext.com/articles/approximation-property).\nA theorem attributed to [Mazur](https://bliptext.com/articles/stanis-aw-mazur) asserts that every infinite-dimensional Banach space V contains a basic sequence, i.e., there is an infinite-dimensional subspace of V that has a Schauder basis.\nThe **basis problem** is the question asked by Banach, whether every separable Banach space has a Schauder basis.\nThis was negatively answered by [Per Enflo](https://bliptext.com/articles/per-enflo) who constructed a separable Banach space failing the approximation property, thus a space without a Schauder basis.\n\n# Examples\nThe standard unit vector bases of c0, and of [ℓp](https://bliptext.com/articles/sequence-space) for 1 ≤ p < ∞, are monotone Schauder bases.\nIn this **unit vector basis** {bn}, the vector bn in or in is the scalar sequence [bn, j]j where all coordinates bn, j are 0, except the nth coordinate:\nwhere δn, j is the [Kronecker delta](https://bliptext.com/articles/kronecker-delta).\nThe space ℓ∞ is not separable, and therefore has no Schauder basis.\nEvery [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) in a separable [Hilbert space](https://bliptext.com/articles/hilbert-space) is a Schauder basis.\nEvery countable orthonormal basis is equivalent to the standard unit vector basis in ℓ2.\nThe [Haar system](https://bliptext.com/articles/haar-wavelet) is an example of a basis for Lp([0, 1]), when 1 ≤ p < ∞.\nWhen 1 < p < ∞, another example is the trigonometric system defined below.\nThe Banach space C([0, 1]) of continuous functions on the interval [0, 1], with the [supremum norm](https://bliptext.com/articles/supremum-norm), admits a Schauder basis.\nThe [Faber–Schauder system](https://bliptext.com/articles/haar-wavelet) is the most commonly used Schauder basis for C([0, 1]).\nSeveral bases for classical spaces were discovered before Banach's book appeared, but some other cases remained open for a long time.\nFor example, the question of whether the [disk algebra](https://bliptext.com/articles/disk-algebra) A(D) has a Schauder basis remained open for more than forty years, until Bočkarev showed in 1974 that a basis constructed from the [Franklin system](https://bliptext.com/articles/haar-wavelet) exists in A(D).\nOne can also prove that the periodic Franklin system is a basis for a Banach space Ar isomorphic to A(D).\n**T**his space Ar consists of all complex continuous functions on the unit circle **T** whose [conjugate function](https://bliptext.com/articles/harmonic-conjugate) is also continuous.\nThe Franklin system is another Schauder basis for C([0, 1]),\nand it is a Schauder basis in Lp([0, 1]) when 1 ≤ p < ∞.\nSystems derived from the Franklin system give bases in the space C1([0, 1]2) of [differentiable](https://bliptext.com/articles/differentiable-function) functions on the unit square.\nThe existence of a Schauder basis in C1([0, 1]2) was a question from Banach's book.\n\n## Relation to Fourier series\nLet {xn} be, in the real case, the sequence of functions\nor, in the complex case,\nThe sequence {xn} is called the **trigonometric system**.\nIt is a Schauder basis for the space Lp([0, 2π]) for any p such that 1 < p < ∞.\nFor p = 2, this is the content of the [Riesz–Fischer theorem](https://bliptext.com/articles/riesz-fischer-theorem), and for p ≠ 2, it is a consequence of the boundedness on the space Lp([0, 2π]) of the [Hilbert transform on the circle](https://bliptext.com/articles/hilbert-transform).\nIt follows from this boundedness that the projections PN defined by\nare uniformly bounded on Lp([0, 2π]) when 1 < p < ∞.\nThis family of maps {PN} is [equicontinuous](https://bliptext.com/articles/equicontinuity) and tends to the identity on the dense subset consisting of [trigonometric polynomials](https://bliptext.com/articles/trigonometric-polynomial).\nIt follows that PNf tends to f in Lp-norm for every f ∈ Lp([0, 2π]).\nIn other words, {xn} is a Schauder basis of Lp([0, 2π]).\nHowever, the set {xn} is not a Schauder basis for L1([0, 2π]).\nThis means that there are functions in L1 whose Fourier series does not converge in the L1 norm, or equivalently, that the projections PN are not uniformly bounded in L1-norm.\nAlso, the set {xn} is not a Schauder basis for C([0, 2π]).\n\n## Bases for spaces of operators\nThe space K(ℓ2) of [compact operators](https://bliptext.com/articles/compact-operator) on the Hilbert space ℓ2 has a Schauder basis.\nFor every x, y in ℓ2, let x ⊗ y denote the [rank one](https://bliptext.com/articles/rank-linear-algebra) operator v ∈ ℓ2 → <v, x > y. If {en}n ≥ 1 is the standard orthonormal basis of ℓ2, a basis for K(ℓ2) is given by the sequence\nFor every n, the sequence consisting of the n2 first vectors in this basis is a suitable ordering of the family {ej ⊗ ek}, for 1 ≤ j, k ≤ n.\nThe preceding result can be generalized: a Banach space X with a basis has the [approximation property](https://bliptext.com/articles/approximation-property), so the space K(X) of compact operators on X is isometrically isomorphic to the [injective tensor product](https://bliptext.com/articles/banach-space)\nIf X is a Banach space with a Schauder basis {en}n ≥ 1 such that the biorthogonal functionals are a basis of the dual, that is to say, a Banach space with a [shrinking basis](https://bliptext.com/articles/schauder-basis), then the space K(X) admits a basis formed by the rank one operators e*j ⊗ ek : v → e*j(v) ek, with the same ordering as before.\nThis applies in particular to every [reflexive](https://bliptext.com/articles/reflexive-space) Banach space X with a Schauder basis\nOn the other hand, the space B(ℓ2) has no basis, since it is non-separable.\nMoreover, B(ℓ2) does not have the approximation property.\n\n# Unconditionality\nA Schauder basis {bn} is **unconditional** if whenever the series converges, it converges\n[unconditionally](https://bliptext.com/articles/unconditional-convergence).\nFor a Schauder basis {bn}, this is equivalent to the existence of a constant C such that\nfor all natural numbers n, all scalar coefficients {αk} and all signs.\nUnconditionality is an important property since it allows one to forget about the order of summation.\nA Schauder basis is **symmetric** if it is unconditional and uniformly equivalent to all its [permutations](https://bliptext.com/articles/permutation): there exists a constant C such that for every natural number n, every permutation π of the set {0, 1, ..., n}, all scalar coefficients {αk} and all signs {εk},\nThe standard bases of the [sequence spaces](https://bliptext.com/articles/sequence-space) c0 and ℓp for 1 ≤ p < ∞, as well as every orthonormal basis in a Hilbert space, are unconditional.\nThese bases are also symmetric.\nThe trigonometric system is not an unconditional basis in Lp, except for p = 2.\nThe Haar system is an unconditional basis in Lp for any 1 < p < ∞.\nThe space L1([0, 1]) has no unconditional basis.\nA natural question is whether every infinite-dimensional Banach space has an infinite-dimensional subspace with an unconditional basis.\nThis was solved negatively by [Timothy Gowers](https://bliptext.com/articles/timothy-gowers) and [Bernard Maurey](https://bliptext.com/articles/bernard-maurey) in 1992.\n\n# Schauder bases and duality\nA basis {en}n≥0 of a Banach space X is **boundedly complete** if for every sequence {an}n≥0 of scalars such that the partial sums\nare bounded in X, the sequence {Vn} converges in X.\nThe unit vector basis for ℓp, 1 ≤ p < ∞, is boundedly complete.\nHowever, the unit vector basis is not boundedly complete in c0.\nIndeed, if an = 1 for every n, then\nfor every n, but the sequence {Vn} is not convergent in c0, since ||Vn+1 − Vn|| = 1 for every n.\nA space X with a boundedly complete basis {en}n≥0 is [isomorphic](https://bliptext.com/articles/banach-space) to a dual space, namely, the space X is [isomorphic](https://bliptext.com/articles/banach-space) to the dual of the closed linear span in the dual X ′ of the biorthogonal functionals associated to the basis {en}.\nA basis {en}n≥0 of X is **shrinking** if for every bounded linear functional f on X, the sequence of non-negative numbers\ntends to 0 when n → ∞, where Fn is the linear span of the basis vectors em for m ≥ n.\nThe unit vector basis for ℓp, 1 < p < ∞, or for c0, is shrinking.\nIt is not shrinking in ℓ1**:** if f is the bounded linear functional on ℓ1 given by then φn ≥ f(en) = 1 for every n.\nA basis [en]n ≥ 0 of X is shrinking if and only if the biorthogonal functionals [e*n]n ≥ 0 form a basis of the dual X ′.\n[Robert C. James](https://bliptext.com/articles/robert-c-james) characterized reflexivity in Banach spaces with basis: the space X with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.\nJames also proved that a space with an unconditional basis is non-reflexive if and only if it contains a subspace isomorphic to c0 or ℓ1.\n\n# Related concepts\nA [Hamel basis](https://bliptext.com/articles/hamel-basis) is a subset B of a vector space V such that every element v ∈ V can uniquely be written as\nwith αb ∈ F, with the extra condition that the set\nis finite.\nThis property makes the Hamel basis unwieldy for infinite-dimensional Banach spaces; as a Hamel basis for an infinite-dimensional Banach space has to be [uncountable](https://bliptext.com/articles/uncountable).\n(Every finite-dimensional subspace of an infinite-dimensional Banach space X has empty [interior](https://bliptext.com/articles/interior-topology), and is [nowhere dense](https://bliptext.com/articles/nowhere-dense) in X. It then follows from the [Baire category theorem](https://bliptext.com/articles/baire-category-theorem) that a countable union of bases of these finite-dimensional subspaces cannot serve as a basis. )","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T06:55:23.630Z","updated_at":"2025-01-26T06:55:23.630Z","search_vector":"'basi':2 'schauder':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **Schauder [basis](https://bliptext.com/articles/basis-linear-algebra)** or **countable [basis](https://bliptext.com/articles/basis-linear-algebra)** is similar to the usual ([Hamel](https://bliptext.com/articles/hamel-[basis](https://bliptext.com/articles/basis-linear-algebra))) [basis](https://bliptext.com/articles/basis-linear-algebra) of a [vector space](https://bliptext.com/articles/vector-space); the difference is that [Hamel](https://bliptext.com/articles/hamel-[basis](https://bliptext.com/articles/basis-linear-algebra)) bases use [linear combinations](https://bliptext.com/articles/linear-combination) that are finite sums, while for Schauder bases they may be infinite sums.\nThis makes Schauder bases more suitable for the analysis of infinite-dimensional [topological vector spaces](https://bliptext.com/articles/topological-vector-space) including [Banach spaces](https://bliptext.com/articles/banach-space).\nSchauder bases were described by [Juliusz Schauder](https://bliptext.com/articles/juliusz-schauder) in 1927, although such bases were discussed earlier.\nFor example, the [Haar basis](https://bliptext.com/articles/haar-basis) was given in 1909, and [Georg Faber](https://bliptext.com/articles/georg-faber) discussed in 1910 a basis for [continuous functions](https://bliptext.com/articles/continuous-function) on an [interval](https://bliptext.com/articles/interval-mathematics), sometimes called a **Faber–Schauder system**.\n\n# Definitions\nLet V denote a [topological vector space](https://bliptext.com/articles/topological-vector-space) over the [field](https://bliptext.com/articles/field-mathematics) F.\nA **Schauder basis** is a [sequence](https://bliptext.com/articles/sequence) {bn} of elements of V such that for every element v ∈ V there exists a unique [sequence](https://bliptext.com/articles/sequence) {αn} of scalars in F so that The convergence of the infinite sum is implicitly that of the ambient topology, i.e., but can be reduced to only [weak convergence](https://bliptext.com/articles/weak-topology) in a [normed vector space](https://bliptext.com/articles/normed-vector-space) (such as a [Banach space](https://bliptext.com/articles/banach-space)).\nUnlike a [Hamel basis](https://bliptext.com/articles/hamel-basis), the elements of the basis must be ordered, since the series may not converge [unconditionally](https://bliptext.com/articles/unconditional-convergence).\nNote that some authors define Schauder bases to be countable (as above), while others use the term to include uncountable bases.\nIn either case, the sums themselves always are countable.\nAn uncountable Schauder basis is a [linearly ordered set](https://bliptext.com/articles/linearly-ordered-set) rather than a sequence, and each sum inherits the order of its terms from this linear ordering.\nThey can and do arise in practice.\nAs an example, a [separable](https://bliptext.com/articles/separable-space) [Hilbert space](https://bliptext.com/articles/hilbert-space) can only have a countable Schauder basis, but a non-[separable](https://bliptext.com/articles/separable-space) [Hilbert space](https://bliptext.com/articles/hilbert-space) may have an uncountable one.\nThough the definition above technically does not require a normed space, a norm is necessary to say almost anything useful about Schauder bases.\nThe results below assume the existence of a norm.\nA Schauder basis {bn}n ≥ 0 is said to be **normalized** when all the basis vectors have norm 1 in the Banach space V.\nA sequence {xn}n ≥ 0 in V is a **basic sequence** if it is a Schauder basis of its [closed linear span](https://bliptext.com/articles/linear-span).\nTwo Schauder bases, {bn} in V and {cn} in W, are said to be **equivalent** if there exist two constants c > 0 and C such that for every [natural number](https://bliptext.com/articles/natural-number) N ≥ 0 and all sequences {αn} of scalars,\nA family of vectors in V is **total** if its [linear span](https://bliptext.com/articles/linear-span) (the [set](https://bliptext.com/articles/set-mathematics) of finite linear combinations) is [dense](https://bliptext.com/articles/dense-subset) in V.\nIf V is a [Hilbert space](https://bliptext.com/articles/hilbert-space), an **[orthogonal basis](https://bliptext.com/articles/orthogonal-basis)** is a total [subset](https://bliptext.com/articles/subset) B of V such that elements in B are nonzero and pairwise orthogonal.\nFurther, when each element in B has norm 1, then B is an **[orthonormal basis](https://bliptext.com/articles/orthonormal-basis)** of V.\n\n# Properties\nLet {bn} be a Schauder basis of a Banach space V over **F** = **R** or **C**.\nIt is a subtle consequence of the [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-functional-analysis) that the linear mappings {Pn} defined by\nare uniformly bounded by some constant C.\nWhen, the basis is called a **monotone** basis.\nThe maps {Pn} are the **[basis projections](https://bliptext.com/articles/projection-linear-algebra)**.\nLet {b*n} denote the **coordinate functionals**, where b*n assigns to every vector v in V the coordinate αn of v in the above expansion.\nEach b*n is a bounded linear functional on V.\nIndeed, for every vector v in V,\nThese functionals {b*n} are called **biorthogonal functionals** associated to the basis {bn}.\nWhen the basis {bn} is normalized, the coordinate functionals {b*n} have norm ≤ 2C in the [continuous dual](https://bliptext.com/articles/dual-space) V ′ of V.\nA Banach space with a Schauder basis is necessarily [separable](https://bliptext.com/articles/separable-space), but the converse is false.\nSince every vector v in a Banach space V with a Schauder basis is the limit of Pn(v), with Pn of finite rank and uniformly bounded, such a space V satisfies the [bounded approximation property](https://bliptext.com/articles/approximation-property).\nA theorem attributed to [Mazur](https://bliptext.com/articles/stanis-aw-mazur) asserts that every infinite-dimensional Banach space V contains a basic sequence, i.e., there is an infinite-dimensional subspace of V that has a Schauder basis.\nThe **basis problem** is the question asked by Banach, whether every separable Banach space has a Schauder basis.\nThis was negatively answered by [Per Enflo](https://bliptext.com/articles/per-enflo) who constructed a separable Banach space failing the approximation property, thus a space without a Schauder basis.\n\n# Examples\nThe standard unit vector bases of c0, and of [ℓp](https://bliptext.com/articles/sequence-space) for 1 ≤ p < ∞, are monotone Schauder bases.\nIn this **unit vector basis** {bn}, the vector bn in or in is the scalar sequence [bn, j]j where all coordinates bn, j are 0, except the nth coordinate:\nwhere δn, j is the [Kronecker delta](https://bliptext.com/articles/kronecker-delta).\nThe space ℓ∞ is not separable, and therefore has no Schauder basis.\nEvery [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) in a separable [Hilbert space](https://bliptext.com/articles/hilbert-space) is a Schauder basis.\nEvery countable orthonormal basis is equivalent to the standard unit vector basis in ℓ2.\nThe [Haar system](https://bliptext.com/articles/haar-wavelet) is an example of a basis for Lp([0, 1]), when 1 ≤ p < ∞.\nWhen 1 < p < ∞, another example is the trigonometric system defined below.\nThe Banach space C([0, 1]) of continuous functions on the interval [0, 1], with the [supremum norm](https://bliptext.com/articles/supremum-norm), admits a Schauder basis.\nThe [Faber–Schauder system](https://bliptext.com/articles/haar-wavelet) is the most commonly used Schauder basis for C([0, 1]).\nSeveral bases for classical spaces were discovered before Banach's book appeared, but some other cases remained open for a long time.\nFor example, the question of whether the [disk algebra](https://bliptext.com/articles/disk-algebra) A(D) has a Schauder basis remained open for more than forty years, until Bočkarev showed in 1974 that a basis constructed from the [Franklin system](https://bliptext.com/articles/haar-wavelet) exists in A(D).\nOne can also prove that the periodic Franklin system is a basis for a Banach space Ar isomorphic to A(D).\n**T**his space Ar consists of all complex continuous functions on the unit circle **T** whose [conjugate function](https://bliptext.com/articles/harmonic-conjugate) is also continuous.\nThe Franklin system is another Schauder basis for C([0, 1]),\nand it is a Schauder basis in Lp([0, 1]) when 1 ≤ p < ∞.\nSystems derived from the Franklin system give bases in the space C1([0, 1]2) of [differentiable](https://bliptext.com/articles/differentiable-function) functions on the unit square.\nThe existence of a Schauder basis in C1([0, 1]2) was a question from Banach's book.\n\n## Relation to Fourier series\nLet {xn} be, in the real case, the sequence of functions\nor, in the complex case,\nThe sequence {xn} is called the **trigonometric system**.\nIt is a Schauder basis for the space Lp([0, 2π]) for any p such that 1 < p < ∞.\nFor p = 2, this is the content of the [Riesz–Fischer theorem](https://bliptext.com/articles/riesz-fischer-theorem), and for p ≠ 2, it is a consequence of the boundedness on the space Lp([0, 2π]) of the [Hilbert transform on the circle](https://bliptext.com/articles/hilbert-transform).\nIt follows from this boundedness that the projections PN defined by\nare uniformly bounded on Lp([0, 2π]) when 1 < p < ∞.\nThis family of maps {PN} is [equicontinuous](https://bliptext.com/articles/equicontinuity) and tends to the identity on the dense subset consisting of [trigonometric polynomials](https://bliptext.com/articles/trigonometric-polynomial).\nIt follows that PNf tends to f in Lp-norm for every f ∈ Lp([0, 2π]).\nIn other words, {xn} is a Schauder basis of Lp([0, 2π]).\nHowever, the set {xn} is not a Schauder basis for L1([0, 2π]).\nThis means that there are functions in L1 whose Fourier series does not converge in the L1 norm, or equivalently, that the projections PN are not uniformly bounded in L1-norm.\nAlso, the set {xn} is not a Schauder basis for C([0, 2π]).\n\n## Bases for spaces of operators\nThe space K(ℓ2) of [compact operators](https://bliptext.com/articles/compact-operator) on the Hilbert space ℓ2 has a Schauder basis.\nFor every x, y in ℓ2, let x ⊗ y denote the [rank one](https://bliptext.com/articles/rank-linear-algebra) operator v ∈ ℓ2 → <v, x > y. If {en}n ≥ 1 is the standard orthonormal basis of ℓ2, a basis for K(ℓ2) is given by the sequence\nFor every n, the sequence consisting of the n2 first vectors in this basis is a suitable ordering of the family {ej ⊗ ek}, for 1 ≤ j, k ≤ n.\nThe preceding result can be generalized: a Banach space X with a basis has the [approximation property](https://bliptext.com/articles/approximation-property), so the space K(X) of compact operators on X is isometrically isomorphic to the [injective tensor product](https://bliptext.com/articles/banach-space)\nIf X is a Banach space with a Schauder basis {en}n ≥ 1 such that the biorthogonal functionals are a basis of the dual, that is to say, a Banach space with a [shrinking basis](https://bliptext.com/articles/schauder-basis), then the space K(X) admits a basis formed by the rank one operators e*j ⊗ ek : v → e*j(v) ek, with the same ordering as before.\nThis applies in particular to every [reflexive](https://bliptext.com/articles/reflexive-space) Banach space X with a Schauder basis\nOn the other hand, the space B(ℓ2) has no basis, since it is non-separable.\nMoreover, B(ℓ2) does not have the approximation property.\n\n# Unconditionality\nA Schauder basis {bn} is **unconditional** if whenever the series converges, it converges\n[unconditionally](https://bliptext.com/articles/unconditional-convergence).\nFor a Schauder basis {bn}, this is equivalent to the existence of a constant C such that\nfor all natural numbers n, all scalar coefficients {αk} and all signs.\nUnconditionality is an important property since it allows one to forget about the order of summation.\nA Schauder basis is **symmetric** if it is unconditional and uniformly equivalent to all its [permutations](https://bliptext.com/articles/permutation): there exists a constant C such that for every natural number n, every permutation π of the set {0, 1, ..., n}, all scalar coefficients {αk} and all signs {εk},\nThe standard bases of the [sequence spaces](https://bliptext.com/articles/sequence-space) c0 and ℓp for 1 ≤ p < ∞, as well as every orthonormal basis in a Hilbert space, are unconditional.\nThese bases are also symmetric.\nThe trigonometric system is not an unconditional basis in Lp, except for p = 2.\nThe Haar system is an unconditional basis in Lp for any 1 < p < ∞.\nThe space L1([0, 1]) has no unconditional basis.\nA natural question is whether every infinite-dimensional Banach space has an infinite-dimensional subspace with an unconditional basis.\nThis was solved negatively by [Timothy Gowers](https://bliptext.com/articles/timothy-gowers) and [Bernard Maurey](https://bliptext.com/articles/bernard-maurey) in 1992.\n\n# Schauder bases and duality\nA basis {en}n≥0 of a Banach space X is **boundedly complete** if for every sequence {an}n≥0 of scalars such that the partial sums\nare bounded in X, the sequence {Vn} converges in X.\nThe unit vector basis for ℓp, 1 ≤ p < ∞, is boundedly complete.\nHowever, the unit vector basis is not boundedly complete in c0.\nIndeed, if an = 1 for every n, then\nfor every n, but the sequence {Vn} is not convergent in c0, since ||Vn+1 − Vn|| = 1 for every n.\nA space X with a boundedly complete basis {en}n≥0 is [isomorphic](https://bliptext.com/articles/banach-space) to a dual space, namely, the space X is [isomorphic](https://bliptext.com/articles/banach-space) to the dual of the closed linear span in the dual X ′ of the biorthogonal functionals associated to the basis {en}.\nA basis {en}n≥0 of X is **shrinking** if for every bounded linear functional f on X, the sequence of non-negative numbers\ntends to 0 when n → ∞, where Fn is the linear span of the basis vectors em for m ≥ n.\nThe unit vector basis for ℓp, 1 < p < ∞, or for c0, is shrinking.\nIt is not shrinking in ℓ1**:** if f is the bounded linear functional on ℓ1 given by then φn ≥ f(en) = 1 for every n.\nA basis [en]n ≥ 0 of X is shrinking if and only if the biorthogonal functionals [e*n]n ≥ 0 form a basis of the dual X ′.\n[Robert C. James](https://bliptext.com/articles/robert-c-james) characterized reflexivity in Banach spaces with basis: the space X with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.\nJames also proved that a space with an unconditional basis is non-reflexive if and only if it contains a subspace isomorphic to c0 or ℓ1.\n\n# Related concepts\nA [Hamel basis](https://bliptext.com/articles/hamel-basis) is a subset B of a vector space V such that every element v ∈ V can uniquely be written as\nwith αb ∈ F, with the extra condition that the set\nis finite.\nThis property makes the Hamel basis unwieldy for infinite-dimensional Banach spaces; as a Hamel basis for an infinite-dimensional Banach space has to be [uncountable](https://bliptext.com/articles/uncountable).\n(Every finite-dimensional subspace of an infinite-dimensional Banach space X has empty [interior](https://bliptext.com/articles/interior-topology), and is [nowhere dense](https://bliptext.com/articles/nowhere-dense) in X. It then follows from the [Baire category theorem](https://bliptext.com/articles/baire-category-theorem) that a countable union of bases of these finite-dimensional subspaces cannot serve as a basis. )","revisionCount":0},"hasMore":false,"nextCursor":null}

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