Absolute value in an ordered field

Depends on

• Ordered field

Used by

• Finite sign enumeration for real phase retrieval Algorithm
• Finite-budget adaptive Runge–Kutta controller Algorithm
• Outward-bounded floating determinant filter Algorithm
• 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
• ℝ((t⁻¹)) has the nested interval property for lengths tending to 0 Corollary
• Uniqueness, comparison and order preservation of entropy solutions Corollary
• {q ∈ ℚ : q ≥ 0, q² < 2} is closed and bounded in ℚ and is not compact Counterexample
• ‖·‖₁ on ℝ² violates the parallelogram law, so no symmetric bilinear form induces it Counterexample
• A function that is not Riemann integrable although | f| is Counterexample
• A null sequence times an unbounded sequence need not be null: (k+1)⁻¹·(k+1) = 1 Counterexample
• A sequence with limsup = +∞: the greatest subsequential limit exists only in ℝ̄ Counterexample
• An R-only proximal coordinate rate fails for nonsmooth penalties Counterexample
• In {0} ∪ [1,2] with the metric of ℝ, the closure of B(0,1) = {0} is {0} while the closed ball is {0,1} Counterexample
• Minkowski fails for p = 1/2: the hypothesis p ≥ 1 is load-bearing Counterexample
• On (0,∞) the metrics |x-y| and |1/x - 1/y| have the same topology and are not uniformly equivalent Counterexample
• On ℝ the metrics |x-y| and min(|x-y|,1) are uniformly but not Lipschitz equivalent Counterexample
• Refuted: a pointwise bounded family of continuous functions is equicontinuous. The spikes are bounded by 1 everywhere and are not equicontinuous at 0 Counterexample
• Refuted: C(X,Y) is closed in the topology of pointwise convergence. The ramps on [0,1] converge pointwise to a discontinuous limit Counterexample
• Refuted: convergence uniformly on every compact subset of ℝ implies uniform convergence. The maps x ↦ x/(n+1) separate the two Counterexample
• The sign function is Riemann integrable on [-1,1] and has no primitive there Counterexample
• The squeeze theorem needs the two bounds to share their limit Counterexample
• The sum of two divergent sequences can converge: k + (-k) Counterexample
• xₖ = 1 + (-1)ᵏ, yₖ = 1 + (-1)ᵏ⁺¹ give limsup(xₖ yₖ) = 0 < 4 Counterexample
• ℤ and {n + 1/n : n ≥ 2} are disjoint closed subsets of ℝ at distance 0, so the set-to-set distance is not a metric Counterexample
• A norm on a real vector space, the induced metric, and the dictionary with the metric axioms Definition
• Absolute and conditional convergence of improper integrals Definition
• Absolute continuity on a compact interval Definition
• Absolutely convergent and conditionally convergent series, and the general starting index Definition
• Bounded variation and total variation on an interval Definition
• Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ Definition
• For n≥1, the determinant over a commutative ring by the Leibniz formula, and |det A| for a real matrix Definition
• Forward and central difference stencils with bounded evaluation errors Definition
• Kruzhkov entropy solutions Definition
• Normwise and componentwise backward error for an approximate linear-system solution Definition
• Permutation matrices, partial pivoting, and the pivot-growth factor Definition
• Sequences, convergence, Cauchyness, monotonicity, boundedness and closed intervals in an arbitrary ordered field Definition
• The complex-step derivative estimator and its component-error model Definition
• The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices Definition
• The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞ Definition
• The standard relative floating-point model and unit roundoff Definition

…and 99 more results.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• T. Tao, Analysis I, 3rd ed. (standard reference, not scraped)
• Dartmouth College analysis notes: Absolute value (standard reference, not scraped)