Bessel inequality

E825431

inequality in inner product spaces mathematical theorem result in functional analysis

Bessel inequality is a fundamental result in functional analysis that bounds the sum of squared Fourier coefficients of a vector in an inner product space by the square of its norm.

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All labels observed (1)

Label	Occurrences
Bessel inequality canonical	2

Statements (47)

Predicate	Object
instanceOf	inequality in inner product spaces ⓘ mathematical theorem ⓘ result in functional analysis ⓘ
appliesTo	Hilbert space NERFINISHED ⓘ inner product space ⓘ
assumes	orthonormality of the system (e_n) ⓘ
category	Hilbert space inequality NERFINISHED ⓘ inequality involving inner products ⓘ
conclusion	series of squared Fourier coefficients is convergent and bounded by \|\|x\|\|^2 ⓘ
doesNotRequire	completeness of the orthonormal system ⓘ
domainRestriction	orthonormal sequence (e_n) in an inner product space ⓘ
equalityCondition	orthonormal system is complete (Parseval identity holds) ⓘ
field	Fourier analysis ⓘ Hilbert space theory ⓘ functional analysis ⓘ
generalizationOf	Pythagorean theorem for infinite orthogonal expansions NERFINISHED ⓘ
holdsIn	complex inner product spaces ⓘ real inner product spaces ⓘ
implies	Fourier coefficients of x are square-summable ⓘ map x ↦ (⟨x,e_n⟩) is bounded from the space into ℓ² ⓘ
involvesConcept	Fourier coefficients ⓘ Parseval identity NERFINISHED ⓘ inner product ⓘ norm ⓘ orthonormal sequence ⓘ orthonormal system ⓘ series expansion ⓘ squared norm ⓘ
logicalForm	for all x and all orthonormal sequences (e_n), sum \|⟨x,e_n⟩\|^2 ≤ \|\|x\|\|^2 ⓘ
mathematicalArea	analysis ⓘ operator theory ⓘ
namedAfter	Friedrich Bessel NERFINISHED ⓘ
relatedTo	Cauchy–Schwarz inequality NERFINISHED ⓘ Parseval theorem NERFINISHED ⓘ Riesz–Fischer theorem NERFINISHED ⓘ
statementForm	sum \|⟨x,e_n⟩\|^2 ≤ \|\|x\|\|^2 ⓘ
typeOfBound	upper bound on energy of Fourier coefficients ⓘ
usedFor	bounding truncation error in Fourier series ⓘ establishing completeness criteria for orthonormal systems ⓘ proving Parseval identity ⓘ stability estimates in Hilbert space expansions ⓘ
usedIn	Fourier series theory NERFINISHED ⓘ Fourier transform theory NERFINISHED ⓘ approximation theory ⓘ signal processing (theoretical foundations) ⓘ spectral theory of operators ⓘ
variable	vector x in an inner product space ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Cauchy–Schwarz inequality → relatedTo → Bessel inequality ⓘ

Riesz–Fischer theorem → relatedTo → Bessel inequality ⓘ