Hölder inequality
E87726
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
Observed surface forms (5)
| Surface form | Occurrences |
|---|---|
| Cauchy–Schwarz inequality | 1 |
| continuous Hölder inequality | 1 |
| discrete Hölder inequality | 1 |
| multilinear Hölder inequality | 1 |
| weighted Hölder inequality | 1 |
Statements (55)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
theorem in analysis ⓘ |
| appliesTo |
integrals of products
ⓘ
measurable functions ⓘ sequences ⓘ sums of products ⓘ |
| condition |
1 ≤ p ≤ ∞
ⓘ
1/p + 1/q = 1 ⓘ |
| domain |
Lp spaces
ⓘ
finite measure spaces ⓘ sigma-finite measure spaces ⓘ |
| equalityCondition |
for sequences, |aᵢ|^p and |bᵢ|^q are proportional
ⓘ
|f|^p and |g|^q are proportional almost everywhere ⓘ |
| field |
functional analysis
ⓘ
harmonic analysis ⓘ mathematical analysis ⓘ measure theory ⓘ probability theory ⓘ |
| generalizes | Cauchy–Schwarz inequality ⓘ |
| hasVariant |
Hölder inequality
self-linksurface differs
ⓘ
surface form:
continuous Hölder inequality
Hölder inequality self-linksurface differs ⓘ
surface form:
discrete Hölder inequality
generalized Hölder inequality ⓘ Hölder inequality self-linksurface differs ⓘ
surface form:
multilinear Hölder inequality
Hölder inequality self-linksurface differs ⓘ
surface form:
weighted Hölder inequality
|
| implies |
Cauchy–Schwarz inequality
ⓘ
Minkowski inequality ⓘ |
| involves |
conjugate exponents
ⓘ
p-norm ⓘ q-norm ⓘ |
| mathematicalDomain |
complex analysis
ⓘ
functional spaces theory ⓘ real analysis ⓘ |
| namedAfter | Otto Hölder NERFINISHED ⓘ |
| relatedTo |
Jensen inequality
ⓘ
Minkowski inequality ⓘ Young inequality ⓘ triangle inequality in Lᵖ ⓘ |
| statement |
For conjugate exponents p and q, ||fg||₁ ≤ ||f||_p ||g||_q
ⓘ
For measurable f in Lᵖ and g in Lᵠ, ∫|fg| ≤ ||f||_p ||g||_q ⓘ For sequences (aᵢ) in ℓᵖ and (bᵢ) in ℓᵠ, Σ|aᵢ bᵢ| ≤ ||a||_p ||b||_q ⓘ |
| usedIn |
Banach space theory
ⓘ
Fourier analysis ⓘ Lebesgue integration ⓘ Lp spaces ⓘ functional estimation ⓘ interpolation theory ⓘ partial differential equations ⓘ probability inequalities ⓘ statistics ⓘ |
| usedToShow |
Lp is a Banach space
ⓘ
Young inequality for convolutions ⓘ absolute convergence of integrals ⓘ boundedness of convolution operators ⓘ boundedness of linear functionals on Lᵖ ⓘ duality between Lᵖ and Lᵠ ⓘ |
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
discrete Hölder inequality
this entity surface form:
continuous Hölder inequality
this entity surface form:
weighted Hölder inequality
this entity surface form:
multilinear Hölder inequality
this entity surface form:
Cauchy–Schwarz inequality