Ordered field

Depends on

• Field

Used by

• Bias-corrected Adam update Algorithm
• Bias-corrected AMSGrad update Algorithm
• Certified complex-polynomial root isolation in exact algebraic arithmetic Algorithm
• Finite sign enumeration for real phase retrieval Algorithm
• RMSProp with an exponential second-moment estimate Algorithm
• Uncorrected Adam for online convex losses Algorithm
• Uncorrected AMSGrad with a coordinatewise historical maximum Algorithm
• A continuous real function on a compact subset of ℝ is bounded Corollary
• Every continuous function on an interval has a primitive; two primitives differ by a constant; and ∫ₐᵇ f = G(b)-G(a) for any primitive G Corollary
• Every nondegenerate interval of ℝ is uncountable Corollary
• For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε Corollary
• If f,g are integrable on [a,b] then so are | f|, f², fg, max(f,g) and min(f,g), and |∫ₐᵇ f| ≤ ∫ₐᵇ| f| Corollary
• If X is nonempty, some row fibre is at least the average size and some row fibre is at most the average size Corollary
• ℝ((t⁻¹)) does not have the least-upper-bound property; its canonical naturals have no supremum Corollary
• ℝ((t⁻¹)) has the nested interval property for lengths tending to 0 Corollary
• Sobolev maxima and minima form a lattice Corollary
• Stolz-Cesaro, 0/0 form: if bₖ is strictly decreasing to 0, aₖ → 0, and the difference quotient converges, then aₖ/bₖ converges to the same value Corollary
• The Cesaro matrix satisfies the Silverman-Toeplitz conditions, giving a second proof of the Cesaro mean theorem Corollary
• The reverse triangle inequality Corollary
• Under dependent choice, a continuous real-valued map on a closed subspace of a normal space extends to the whole space, and a map into an open interval extends into that same open interval Corollary
• ι(Dₙ) = ι(n) ι(Dₙ₋₁) + (-1)ⁿ for n ≥ 1, and Dₙ = (n-1)(Dₙ₋₁ + Dₙ₋₂) for n ≥ 2 Corollary
• [0,1) is neither open nor closed in ℝ Counterexample
• {0} ∪ [1,2] is closed, has an isolated point, and is not perfect Counterexample
• {q ∈ ℚ : q ≥ 0, q² < 2} is closed and bounded in ℚ and is not compact Counterexample
• ⋂ₖ (-1/k, 1/k) = {0} is not open Counterexample
• 1/4 lies in the Cantor set and is the endpoint of no removed interval, so the endpoints do not exhaust it Counterexample
• A bounded continuous normalized function that is not of positive type Counterexample
• A field homomorphism of ordered fields need not preserve order Counterexample
• A function differentiable on [0,1] whose derivative is unbounded, hence not Riemann integrable Counterexample
• A function that is not Riemann integrable although | f| is Counterexample
• A list of six distinct reals with no strictly increasing sublist of length four and no strictly decreasing sublist of length three Counterexample
• A null sequence times an unbounded sequence need not be null: (k+1)⁻¹·(k+1) = 1 Counterexample
• A relation whose row fibres all differ from the average size, so the averaging principle gives a bound that no fibre meets exactly Counterexample
• A sequence with limsup = +∞: the greatest subsequential limit exists only in ℝ̄ Counterexample
• A summability matrix failing exactly one Silverman-Toeplitz condition and transforming a convergent sequence to a divergent one Counterexample
• A supremum need not belong to its set: sup(0,1) = 1 ∉ (0,1) Counterexample
• A three-set count that drops the triple intersection and returns the wrong answer Counterexample
• aₖ = (-1)ᵏ, bₖ = k have aₖ/bₖ → 0 while the difference quotient oscillates, so Stolz-Cesaro has no converse Counterexample
• AM-GM fails for negative entries: (-1,-4) has geometric mean 2 and arithmetic mean -5/2 Counterexample
• An unbounded set has no supremum: the naturals inside ℝ Counterexample

…and 376 more results.

Dependency tree · one level

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Sources

• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• M. Spivak, Calculus, 4th ed., Ch. 1 (standard reference, not scraped)
• University of Illinois Chicago notes: Ordered field axioms (standard reference, not scraped)