Extended real number line
Description
The extended real number line is the set . It has two new elements, and , that are not real numbers. Its order extends the order of , with for each real number .
Remarks
In , each subset has a supremum and an infimum. For example, and . Some operations extend: for each real . Other expressions, such as , stay undefined. The set is not a field.
Where this leads
120 pages build on this one.
- Arzelà–Ascoli theorem
- Lebesgue number lemma
- Compact iff complete and totally bounded
- Diameter of a set
- Heine–Cantor theorem
- Absolutely convergent series in a Banach space converge
- Complete metric space
- Sequence spaces lp
- Completeness of the space of bounded continuous functions
- Baire category theorem
- Finite-dimensional normed spaces are complete
- Banach space
- Closed subsets of complete spaces are complete
- Completion of a metric space
- Compactness and sequential compactness agree in metric spaces
- Epsilon characterisation of Cauchy sequences
- Completeness via Cauchy sequences
- Epsilon-net characterisation of total boundedness
- Sup norm on bounded continuous functions
- Banach fixed point theorem
- Compact closed unit ball implies finite dimension
- Complete subspaces are closed
- Totally bounded set
- Finitely additive measure
- Stone–Weierstrass theorem
- Fundamental theorem of Γ-convergence
- Köthe–Toeplitz dual
- L2 convergence of Fourier series
- Convergence of the Neumann series for Volterra kernels
- Exponential of a bounded operator
- Series in Banach spaces
- Céa's lemma
- Pettis integral
- Strongly measurable function
- Epigraph
- Hille–Yosida theorem
- Vanishing mean oscillation
- Banach bundle
- Fixed-point iteration
- Measure
- Lax–Milgram theorem
- Neumann series
- Lebesgue covering dimension
- Continuous functional calculus
- Nash–Moser theorem
- Minimum total potential energy principle
- Modulus of continuity
- Banach limit
- Signed measure
- Fenchel conjugate
- Implicit function theorem in Banach spaces
- Dedekind–MacNeille completion
- Dirichlet form
- FK-space
- Bochner integral
- Approximation of a kernel by degenerate kernels
- Complex p-adic numbers
- Continuous function with divergent Fourier series
- de Rham curve
- Bounded sequences ℓ∞
- Sequence space
- Krasner's lemma
- Integral of a non-negative function
- Reduction of a first-kind Volterra equation to the second kind
- Space of convergent sequences
- Liouville–Neumann series
- Dvoretzky–Rogers theorem
- Carathéodory's criterion
- Schauder fixed-point theorem for integral equations
- Content
- Poincaré inequality
- Berkovich space dynamics
- Contraction mapping principle for integral equations
- Density of trigonometric polynomials in continuous functions
- Completion of a valued field
- Spectral radius formula
- Inverse function theorem in Banach spaces
- Backward Euler method
- Compactness of integral operators
- Hankel operator
- Galerkin method
- Compactness of families of harmonic functions
- Premeasure
- Orlicz–Pettis theorem
- Γ-convergence
- Fenchel–Young inequality
- Commutative Gelfand–Naimark theorem
- Fourier basis of L2 of the circle
- Hilbert manifold
- Jentzsch's theorem on positive kernels
- Countable additivity
- Banach–Stone theorem
- Banach–Mazur theorem
- Moore–Osgood theorem
- Mosco convergence
- Noncommutative geometry
- Schauder basis
- Tsirelson space
- F-space
- Ladyzhenskaya–Babuška–Brezzi condition
- Resolvent as Laplace transform of a semigroup
- Banach algebra
- Tame Fréchet space
- Basel problem via Fourier series
- Banach–Lie group
- BK-space
- Convenient vector space
- Outer measure
- Space of null sequences
- Unique solvability of Volterra equations of the second kind
- Fenchel–Moreau theorem
- Semi-continuity
- Dini derivative
- Automorphism group of a bounded domain is a Lie group
- Rellich–Kondrachov theorem
- Oscillation of a function
- Limiting subdifferential
- Semicontinuity
- Montel's theorem
- Essential supremum
Further reading
- Web https://en.wikipedia.org/wiki/Extended_real_number_line, lead section
- Web https://en.wikipedia.org/wiki/Extended_real_number_line, Order and topological properties
About this page
| Tag | 0GSZ7B0CGR |
|---|---|
| URI | m2://0GSZ7B0CGR |
| Aliases | extended-reals |
| Kind | object |
| Writing | W1 stated |
| MSC | 26A03 |
| Same as |
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What this page needs for W2
- section Invariants: a table of the main invariants (0 of 1)
- edge instance_of out (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Used by
- Diameter of a set
- Finitely additive measure
- Fundamental theorem of Γ-convergence
- Epigraph
- Measure
- Modulus of continuity
- Signed measure
- Fenchel conjugate
- Dedekind–MacNeille completion
- Integral of a non-negative function
- Content
- Premeasure
- Γ-convergence
- Fenchel–Young inequality
- Countable additivity
- Mosco convergence
- Outer measure
- Fenchel–Moreau theorem
- Semi-continuity
- Dini derivative
- Oscillation of a function
- Limiting subdifferential
- Semicontinuity
- Essential supremum