Order, lattices, ordered algebraic structures Definition
Set bounded above
Definition
Let be a partially ordered set and let be a subset of . An element of is an upper bound of when for each in . The set is bounded above when it has at least one upper bound in . Similarly, is a lower bound of when for each in . The set is bounded below when it has a lower bound.
Remarks
In , the interval is bounded above. Each number is an upper bound. The set is not bounded above in , by the Archimedean property. An upper bound of does not need to be an element of .
Where this leads
418 pages build on this one.
- Modulus of the conjugate
- Residue at a simple pole
- Derivative of the complex exponential
- Cauchy's integral theorem
- Arguments add under multiplication
- Holomorphic functions on a disc have primitives
- Casorati–Weierstrass theorem
- Chebyshev's inequality
- Riemann mapping theorem
- The real part of a holomorphic function is harmonic
- Residue theorem
- Formula for the modulus
- Arzelà–Ascoli theorem
- Equivalence of norms in finite dimension
- Open mapping theorem
- Lebesgue number lemma
- Entire function
- Uniform limits of holomorphic functions are holomorphic
- Every real number has a decimal expansion
- Maximum modulus principle
- Correlation coefficient
- Uniqueness of the supremum
- Density of the irrationals
- Infimum
- Compact iff complete and totally bounded
- The nth roots of unity
- Order of a zero
- Repeating decimals are rational
- Existence of square roots of nonnegative reals
- Diameter of a set
- Heine–Cantor theorem
- Branch cut
- Existence of Laurent expansions on an annulus
- Decimal expansion
- Distance from a point to a set
- Holomorphic functions are analytic
- Absolutely convergent series in a Banach space converge
- Complete metric space
- Sequence spaces lp
- Picard's great theorem
- Domain in the complex plane
- Intermediate value theorem for connected spaces
- Power series of the complex logarithm
- Completeness of the space of bounded continuous functions
- The reals are Archimedean
- Baire category theorem
- Finite-dimensional normed spaces are complete
- Metrizable spaces are normal
- Annulus
- Ceiling function
- Completeness axiom
- Quantile function
- Connected subsets of the real line are intervals
- Banach space
- Weak law of large numbers
- Goursat's theorem
- Polar form
- Riemann's theorem on removable singularities
- Supremum
- Linear maps on finite-dimensional spaces are continuous
- Kurtosis
- Harmonic conjugate
- Zero of a holomorphic function
- exp(2 pi i/n) is a primitive root of unity
- Complex differentiability
- Rouché's theorem
- Strong law of large numbers
- Holomorphic function
- Cauchy's integral formula
- Irrationality of the square root of 2
- Number of nth roots of unity
- Existence of the infimum
- Liouville's theorem
- 68–95–99.7 rule
- Cauchy's integral formula for derivatives
- Closed subsets of complete spaces are complete
- Inverse of a complex number
- The complex exponential is entire
- Completion of a metric space
- Skewness
- Modulus is multiplicative
- Mean value property
- Residue
- Cauchy's estimates
- Fundamental theorem of algebra
- Isolated zeros theorem
- Compactness and sequential compactness agree in metric spaces
- Modulus of a complex number
- Quadratic formula
- Approximation property of the supremum
- Schwarz lemma
- Epsilon characterisation of Cauchy sequences
- Completeness via Cauchy sequences
- exp(1/z) has an essential singularity at 0
- Principal part
- Epsilon-net characterisation of total boundedness
- Meromorphic function
- Removable singularity
- Sup norm on bounded continuous functions
- Heine–Borel theorem in finite-dimensional normed spaces
- Argument principle
- Holomorphic functions with nonzero derivative are conformal
- Holomorphic functions are infinitely differentiable
- Characterization of the floor
- Branch of the logarithm
- Principal value of the argument
- Hurwitz's theorem
- Density of the rationals
- Cauchy–Riemann equations
- Isolated singularity
- Continuous real function on a compact set attains its bounds
- Square root
- Morera's theorem
- Banach fixed point theorem
- Standard deviation
- Compact closed unit ball implies finite dimension
- Complete subspaces are closed
- Totally bounded set
- There is a natural number n with 1/n below any positive epsilon
- Pole
- Complex logarithm
- The principal logarithm is holomorphic off the branch cut
- Identity theorem
- Irrationality of the square root of a prime
- Order of a pole
- Essential singularity
- Principal branch
- Existence of nth roots of complex numbers
- Power series are holomorphic in the disc of convergence
- Fundamental theorem of contour integration
- Floor function
- The complex conjugate is not holomorphic
- A complex number times its conjugate
- Path-connected spaces are connected
- Operator norm
- Complex power
- Argument of a complex number
- Function with no Laplace transform
- Weierstrass factorization theorem
- Univalent function
- Riesz projector
- Stone–Weierstrass theorem
- Mahler measure
- Arakelyan's theorem
- Conformal invariance of harmonic functions
- Shell theorem
- Ergodic decomposition
- Ordinal arithmetic
- Köthe–Toeplitz dual
- Koksma–Hlawka inequality
- Phragmén–Lindelöf principle for subharmonic functions
- Repair radii and villainy
- Namioka--Phelps
- Nevanlinna characteristic
- Cauchy–Pompeiu formula
- Weierstrass elliptic function
- Weierstrass zeta and sigma functions
- L2 convergence of Fourier series
- Joukowsky transform
- Cheng–Yau gradient estimate
- Schwarzian derivative
- Removable singularities for harmonic functions
- Minimum principle for harmonic functions
- Lyusternik's theorem on tangent spaces
- Convergence of the Neumann series for Volterra kernels
- Exponential of a bounded operator
- Durand–Kerner method
- Hardy's uncertainty principle
- Schwarz–Christoffel mapping
- Series in Banach spaces
- Jensen's formula
- Teichmüller space
- Nevanlinna–Pick interpolation
- Céa's lemma
- Modular lambda function
- Consistency of the maximum likelihood estimator
- Monodromy theorem
- Fodor's lemma
- Artin–Schreier theorem
- Perron's irreducibility criterion
- Elliptic function
- Pettis integral
- Strongly measurable function
- Uniformization theorem
- Brent's method
- Mergelyan's theorem
- High indices theorem
- Stein interpolation theorem
- Kobayashi metric
- Hille–Yosida theorem
- Perron integral
- Wirtinger derivatives
- Runge's theorem
- Carlson's theorem
- Stationary set
- Isoperimetric problem of the calculus of variations
- Harmonic majorant
- Vanishing mean oscillation
- Banach bundle
- Veblen function
- Gamma function as Mellin transform of the exponential
- Analyticity of the Laplace transform in its half-plane
- Splitting circle method
- Fixed-point iteration
- Identity theorem in several variables
- Log-modulus of a holomorphic function is subharmonic
- Bisection method
- Kingman's subadditive ergodic theorem
- Lax–Milgram theorem
- Fekete's lemma
- Holomorphic map between Riemann surfaces
- Cauchy-type integral
- Asymptotic normality of the maximum likelihood estimator
- Neumann series
- Catenary problem
- Maximum modulus principle in several variables
- Weierstrass product
- Lebesgue covering dimension
- Weyl's lemma
- Zeros and poles
- Continuous functional calculus
- Bromwich integral
- Nash–Moser theorem
- Poisson–Jensen formula
- Minimum total potential energy principle
- Modulus of continuity
- Banach limit
- Carathéodory's theorem on boundary extension
- Smoothness and analyticity of harmonic functions
- Green's function and the Riemann map
- Band-limited function
- Landau's constants
- Implicit function theorem in Banach spaces
- Sub-mean value property
- Error of best approximation
- Blaschke product
- Borel–Ritt theorem
- Landau–Kolmogorov inequality
- Euler's elastica
- Kakutani's solution of the Dirichlet problem
- Brennan conjecture
- Legendre duplication formula
- Converse of the mean value property
- Local existence of harmonic conjugates
- Pólya's theorem on integer-valued entire functions
- Space of bounded continuous functions
- Post's inversion formula
- Hadamard gap theorem
- Monte Carlo integration
- Earnshaw's theorem
- Conformal radius
- Nehari's theorem on univalence
- Euler's sine product
- Completeness of the real numbers
- Total variation of a measure
- Nontangential limit
- Repair radii and villainy
- Hartogs's theorem on separate holomorphicity
- Harmonic measure
- FK-space
- Bochner integral
- Superharmonic potential
- Total variation
- Borel–Carathéodory theorem
- Dido's problem
- Nested radical
- Cauchy integral formula on a polydisc
- Approximation of a kernel by degenerate kernels
- Complex p-adic numbers
- Continuous function with divergent Fourier series
- de Rham curve
- Phragmén–Lindelöf principle
- Strip of convergence of the bilateral transform
- Bounded sequences ℓ∞
- Sequence space
- Krasner's lemma
- Integral of a non-negative function
- Kramers–Kronig relations
- Reduction of a first-kind Volterra equation to the second kind
- Space of convergent sequences
- Club set
- Second main theorem of Nevanlinna theory
- Liouville–Neumann series
- Dvoretzky–Rogers theorem
- Legendre transformation
- Nevanlinna theory
- Iterated Laplace transform equals Stieltjes transform
- Quantile regression
- Decreasing rearrangement
- Birkhoff's ergodic theorem
- Diamond principle
- Schauder fixed-point theorem for integral equations
- Poincaré inequality
- Order of a zero or pole
- Berkovich space dynamics
- Wiman–Valiron theory
- Conformal modulus
- Contraction mapping principle for integral equations
- Density of trigonometric polynomials in continuous functions
- Completion of a valued field
- Proper forcing axiom
- Koebe's theorem on the mean value property
- Bloch's theorem
- Spectral radius formula
- Hadamard factorization theorem
- Peetre K-functional
- Spearman's rank correlation coefficient
- Inverse function theorem in Banach spaces
- Backward Euler method
- Nested intervals theorem
- Compactness of integral operators
- Hankel operator
- Perron method
- Galerkin method
- Wiener–Hopf method
- Compactness of families of harmonic functions
- Resurgent function
- Bootstrap
- Gradient estimates for harmonic functions
- Osgood's lemma
- Orlicz–Pettis theorem
- Oseledets theorem
- Commutative Gelfand–Naimark theorem
- Carathéodory metric
- Fourier basis of L2 of the circle
- Brody hyperbolicity
- Schwarz reflection principle
- Hilbert manifold
- Jentzsch's theorem on positive kernels
- Laguerre–Pólya class
- Nörlund–Rice integral
- Littlewood subordination theorem
- Normal function
- Fatou's theorem
- Lagrange multipliers on Banach spaces
- Banach–Stone theorem
- Banach–Mazur theorem
- Ahlfors theory
- Poles of the gamma function
- First main theorem of Nevanlinna theory
- Liouville's theorem for harmonic functions
- Moore–Osgood theorem
- Cauchy–Riemann equations in several variables
- Normal family
- Picard's little theorem
- Liouville's theorems on elliptic functions
- Cauchy–Riemann operator
- Harmonic functions on Riemann surfaces
- Noncommutative geometry
- Schauder basis
- Erdős–Rado theorem
- Tsirelson space
- Denjoy–Carleman–Ahlfors theorem
- F-space
- Ladyzhenskaya–Babuška–Brezzi condition
- Maximum and minimum
- Euler–Gauss limit formula
- Resolvent as Laplace transform of a semigroup
- Liouville's theorem for planar harmonic functions
- Maximum principle for harmonic functions
- Inner function
- Outer function
- Aleph number
- Banach algebra
- Paley–Wiener theorem
- Nevanlinna class
- Hadamard three-circle theorem for subharmonic functions
- Tame Fréchet space
- Lindelöf's maximum principle
- Basel problem via Fourier series
- Hadamard three-circle theorem
- Schwarz–Pick theorem
- Banach–Lie group
- Perron–Wiener–Brelot solution
- BK-space
- Differentiation in the transform variable
- Jury stability criterion
- Pesin theory
- Grunsky matrix
- Exponential type
- Convenient vector space
- Gamma function has no zeros
- Space of null sequences
- Harmonic measure as exit distribution
- Natural boundary
- Reflection formula for the gamma function
- Riemann surface
- Unique solvability of Volterra equations of the second kind
- Hadamard three-lines theorem
- Corona theorem
- Parabolic and hyperbolic Riemann surfaces
- Analytic capacity
- Mellin's inverse formula
- Mittag-Leffler's theorem
- Lindelöf's theorem
- Branch point
- Best approximation
- Hadamard's two-constants theorem
- Disk algebra
- Bôcher's theorem
- Gauss multiplication formula
- de Branges space
- Chebyshev constant
- Reciprocal gamma function
- Automorphism group of a bounded domain is a Lie group
- Circle average
- Rellich–Kondrachov theorem
- Multiplication theorem
- Oscillation of a function
- Order of an entire function
- Euler integral for the gamma function
- Beth number
- Functional equation of the gamma function
- Hartogs lemma for subharmonic functions
- Faber polynomials
- Montel's theorem
- Essential supremum
- Residue calculus for Laplace inversion
Further reading
- Web https://en.wikipedia.org/wiki/Upper_and_lower_bounds, lead section
About this page
| Tag | 0B5H2V4JW8 |
|---|---|
| URI | m2://0B5H2V4JW8 |
| Kind | definition |
| Writing | W0 stub |
| MSC | 06A06 |
| Defines | Set bounded above |
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